172:@0.031627:0.974518:0.062330:0.974518:0.062330:0.956333:0.031627:0.956333:0.000000:0.000000:0.000000
Aplicaciones de la derivada:@0.073089:0.043661:0.316384:0.043661:0.316384:0.024916:0.073089:0.024916:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Ejemplo 2.:@0.293660:0.136836:0.372959:0.136836:0.372959:0.120167:0.293660:0.120167:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
 :@0.372838:0.136836:0.376869:0.136836:0.376869:0.120402:0.372838:0.120402:0.000000
Con base en la gráfica dada en la Figura 2.7, analicemos cómo es la tangente :@0.376556:0.136836:0.931031:0.136836:0.931031:0.120402:0.376556:0.120402:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
a la curva en los puntos donde hay un máximo relativo o un mínimo relativo. Además, :@0.293668:0.153478:0.930466:0.153478:0.930466:0.137044:0.293668:0.137044:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
determinemos dichos valores.:@0.293668:0.170120:0.511158:0.170120:0.511158:0.153686:0.293668:0.153686:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Figura 2.7:@0.581712:0.193279:0.638906:0.193279:0.638906:0.179833:0.581712:0.179833:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
d:@0.642633:0.354762:0.650897:0.354762:0.650897:0.341667:0.642633:0.341667:0.000000
-2:@0.586043:0.402360:0.596540:0.402360:0.596540:0.390767:0.586043:0.390767:0.000000:0.000000
0:@0.589435:0.352352:0.596002:0.352352:0.596002:0.340759:0.589435:0.340759:0.000000
4:@0.589435:0.233404:0.596002:0.233404:0.596002:0.221811:0.589435:0.221811:0.000000
x:@0.731746:0.362170:0.738646:0.362170:0.738646:0.347989:0.731746:0.347989:0.000000
y:@0.578257:0.226384:0.585157:0.226384:0.585157:0.212203:0.578257:0.212203:0.000000
y = x:@0.634178:0.252356:0.662998:0.252356:0.662998:0.238175:0.634178:0.238175:0.000000:0.000000:0.000000:0.000000:0.000000
3 :@0.662997:0.248606:0.672278:0.248606:0.672278:0.237013:0.662997:0.237013:0.000000:0.000000
- 3:@0.672277:0.252356:0.688424:0.252356:0.688424:0.238187:0.672277:0.238187:0.000000:0.000000:0.000000
x:@0.688424:0.252356:0.695324:0.252356:0.695324:0.238175:0.688424:0.238175:0.000000
f’(c) = :@0.530600:0.279415:0.569387:0.279415:0.569387:0.265234:0.530600:0.265234:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
0:@0.569387:0.279415:0.577413:0.279415:0.577413:0.265245:0.569387:0.265245:0.000000
f’(d) = :@0.622968:0.412566:0.663350:0.412566:0.663350:0.398385:0.622968:0.398385:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
0:@0.663350:0.412566:0.671377:0.412566:0.671377:0.398396:0.663350:0.398396:0.000000
2:@0.694686:0.352351:0.701253:0.352351:0.701253:0.340758:0.694686:0.340758:0.000000
c:@0.544953:0.354762:0.551339:0.354762:0.551339:0.341667:0.544953:0.341667:0.000000
-2:@0.492639:0.352351:0.503136:0.352351:0.503136:0.340758:0.492639:0.340758:0.000000:0.000000
2:@0.589430:0.290552:0.595997:0.290552:0.595997:0.278959:0.589430:0.278959:0.000000
En la gráfica de la función podemos observar que existe un máximo relativo en   y un :@0.293660:0.450493:0.930974:0.450493:0.930974:0.434059:0.293660:0.434059:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
c:@0.882435:0.450493:0.890092:0.450493:0.890092:0.434087:0.882435:0.434087:0.000000
mínimo relativo en  . En los puntos máximo y mínimo la tangente es horizontal, puesto :@0.293660:0.467135:0.931262:0.467135:0.931262:0.450701:0.293660:0.450701:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
d:@0.434641:0.467135:0.443900:0.467135:0.443900:0.450729:0.434641:0.450729:0.000000
que la derivada en   y en   son iguales a cero, lo que nos indica que las pendientes de las :@0.293660:0.483777:0.930796:0.483777:0.930796:0.467343:0.293660:0.467343:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
c:@0.429230:0.483777:0.436887:0.483777:0.436887:0.467371:0.429230:0.467371:0.000000
d:@0.473922:0.483777:0.483181:0.483777:0.483181:0.467371:0.473922:0.467371:0.000000
rectas tangentes son nulas.:@0.293642:0.500419:0.489653:0.500419:0.489653:0.483985:0.293642:0.483985:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Verifiquemos que eso es cierto.:@0.293642:0.524203:0.519022:0.524203:0.519022:0.507769:0.293642:0.507769:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Como  ( ) = :@0.293642:0.544422:0.382806:0.544422:0.382806:0.527989:0.293642:0.527989:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f x:@0.342310:0.544422:0.359190:0.544422:0.359190:0.528016:0.342310:0.528016:0.000000:0.000000:0.000000
x:@0.382696:0.544422:0.390151:0.544422:0.390151:0.528016:0.382696:0.528016:0.000000
3 :@0.390121:0.538360:0.397665:0.538360:0.397665:0.528779:0.390121:0.528779:0.000000:0.000000
–:@0.397614:0.545006:0.406818:0.545006:0.406818:0.527352:0.397614:0.527352:0.000000
 3 , entonces  ´( ) = 3:@0.406726:0.544424:0.563464:0.544424:0.563464:0.527990:0.406726:0.527990:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x:@0.419574:0.544424:0.427029:0.544424:0.427029:0.528018:0.419574:0.528018:0.000000
f  x:@0.505370:0.544424:0.530938:0.544424:0.530938:0.528018:0.505370:0.528018:0.000000:0.000000:0.000000:0.000000
x:@0.563353:0.544424:0.570808:0.544424:0.570808:0.528018:0.563353:0.528018:0.000000
2 :@0.570760:0.538360:0.578304:0.538360:0.578304:0.528779:0.570760:0.528779:0.000000:0.000000
–:@0.578253:0.545006:0.587457:0.545006:0.587457:0.527352:0.578253:0.527352:0.000000
 3.:@0.587365:0.544424:0.603434:0.544424:0.603434:0.527990:0.587365:0.527990:0.000000:0.000000:0.000000
Ahora, miramos dónde  ´( )=0, es decir, igualamos la primera derivada a 0, con lo que :@0.293658:0.571786:0.930919:0.571786:0.930919:0.555352:0.293658:0.555352:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f x:@0.468785:0.571786:0.491444:0.571786:0.491444:0.555380:0.468785:0.555380:0.000000:0.000000:0.000000
obtenemos la ecuación3    3 = 0.:@0.293677:0.588428:0.542600:0.588429:0.542600:0.571995:0.293677:0.571994:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x:@0.473184:0.588428:0.480639:0.588428:0.480639:0.572021:0.473184:0.572021:0.000000
2:@0.480549:0.582365:0.485786:0.582365:0.485786:0.572784:0.480549:0.572784:0.000000
–:@0.489697:0.589011:0.498901:0.589011:0.498901:0.571357:0.489697:0.571357:0.000000
Para hallar los puntos críticos solucionamos la ecuación.:@0.293660:0.615791:0.697589:0.615791:0.697589:0.599357:0.293660:0.599357:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
3:@0.293660:0.636011:0.302643:0.636011:0.302643:0.619577:0.293660:0.619577:0.000000
x:@0.302569:0.636011:0.310024:0.636011:0.310024:0.619604:0.302569:0.619604:0.000000
2 :@0.309951:0.629944:0.317495:0.629944:0.317495:0.620363:0.309951:0.620363:0.000000:0.000000
– :@0.317444:0.636591:0.331157:0.636591:0.331157:0.618936:0.317444:0.618936:0.000000:0.000000
3 = 0 :@0.331065:0.636008:0.371727:0.636008:0.371727:0.619574:0.331065:0.619574:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f´(x)=:@0.435963:0.636008:0.475373:0.636008:0.475373:0.619602:0.435963:0.619602:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
0:@0.475336:0.636008:0.484319:0.636008:0.484319:0.619574:0.475336:0.619574:0.000000
3(:@0.293657:0.656228:0.307444:0.656228:0.307444:0.639794:0.293657:0.639794:0.000000:0.000000
x:@0.307352:0.656228:0.314807:0.656228:0.314807:0.639822:0.307352:0.639822:0.000000
2 :@0.314748:0.650160:0.322292:0.650160:0.322292:0.640579:0.314748:0.640579:0.000000:0.000000
– :@0.322240:0.656807:0.335953:0.656807:0.335953:0.639152:0.322240:0.639152:0.000000:0.000000
1) = 0 :@0.335861:0.656224:0.381329:0.656224:0.381329:0.639790:0.335861:0.639790:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Factorizando :@0.435960:0.656224:0.533756:0.656224:0.533756:0.639790:0.435960:0.639790:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
3(:@0.293660:0.676439:0.307447:0.676439:0.307447:0.660005:0.293660:0.660005:0.000000:0.000000
x :@0.307355:0.676439:0.318050:0.676439:0.318050:0.660033:0.307355:0.660033:0.000000:0.000000
–:@0.317995:0.677021:0.327198:0.677021:0.327198:0.659367:0.317995:0.659367:0.000000
 1)( + 1) = 0  Factorizando:@0.327106:0.676439:0.529729:0.676439:0.529729:0.660005:0.327106:0.660005:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x :@0.349545:0.676439:0.360239:0.676439:0.360239:0.660033:0.349545:0.660033:0.000000:0.000000
x:@0.293660:0.703802:0.301115:0.703802:0.301115:0.687396:0.293660:0.687396:0.000000
1 :@0.301132:0.703548:0.307694:0.703548:0.307694:0.695274:0.301132:0.695274:0.000000:0.000000
=  1  :@0.307963:0.703802:0.350336:0.703802:0.350336:0.687368:0.307963:0.687368:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
–:@0.323517:0.704385:0.332720:0.704385:0.332720:0.686730:0.323517:0.686730:0.000000
 x:@0.350870:0.703802:0.362080:0.703802:0.362080:0.687396:0.350870:0.687396:0.000000:0.000000
2 :@0.362098:0.703548:0.368660:0.703548:0.368660:0.695274:0.362098:0.695274:0.000000:0.000000
= 1   son los puntos críticos, que en la gráfica de la Figura 2.7 corresponden :@0.368929:0.703802:0.930939:0.703802:0.930939:0.687368:0.368929:0.687368:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
a   y  .:@0.293662:0.720444:0.341778:0.720444:0.341778:0.704010:0.293662:0.704010:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
c d:@0.305884:0.720444:0.338649:0.720444:0.338649:0.704038:0.305884:0.704038:0.000000:0.000000:0.000000
Como   está definida para todo  , esos son los únicos puntos críticos de  . :@0.293643:0.747806:0.824343:0.747806:0.824343:0.731372:0.293643:0.731372:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f:@0.342312:0.747806:0.347006:0.747806:0.347006:0.731400:0.342312:0.731400:0.000000
x:@0.522077:0.747806:0.529532:0.747806:0.529532:0.731400:0.522077:0.731400:0.000000
f:@0.812489:0.747806:0.817183:0.747806:0.817183:0.731400:0.812489:0.731400:0.000000
Al evaluar   en esos números, encontramos el valor máximo y el mínimo.:@0.293607:0.775168:0.816930:0.775168:0.816930:0.758734:0.293607:0.758734:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f:@0.368598:0.775168:0.373292:0.775168:0.373292:0.758762:0.368598:0.758762:0.000000
f:@0.293607:0.795388:0.298300:0.795388:0.298300:0.778982:0.293607:0.778982:0.000000
( 1) = ( 1):@0.298227:0.795388:0.372334:0.795388:0.372334:0.778954:0.298227:0.778954:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
–:@0.303031:0.795970:0.312235:0.795970:0.312235:0.778316:0.303031:0.778316:0.000000
–:@0.349436:0.795970:0.358639:0.795970:0.358639:0.778316:0.349436:0.778316:0.000000
3  :@0.372294:0.789322:0.382146:0.789322:0.382146:0.779742:0.372294:0.779742:0.000000:0.000000:0.000000
– –:@0.382092:0.795969:0.414102:0.795969:0.414102:0.778315:0.382092:0.778315:0.000000:0.000000:0.000000
3( 1) = 2.   :@0.391204:0.795386:0.478465:0.795386:0.478465:0.778953:0.391204:0.778953:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f:@0.534628:0.795386:0.539322:0.795386:0.539322:0.778980:0.534628:0.778980:0.000000
( 1) = (1):@0.539248:0.795386:0.604281:0.795386:0.604281:0.778953:0.539248:0.778953:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
–:@0.544052:0.795969:0.553256:0.795969:0.553256:0.778315:0.544052:0.778315:0.000000
3   :@0.604150:0.789322:0.623105:0.789322:0.623105:0.779742:0.604150:0.779742:0.000000:0.000000:0.000000:0.000000
–:@0.611644:0.795969:0.620847:0.795969:0.620847:0.778315:0.611644:0.778315:0.000000
3(1) =  2.:@0.623060:0.795386:0.690467:0.795386:0.690467:0.778953:0.623060:0.778953:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
–:@0.669243:0.795969:0.678447:0.795969:0.678447:0.778315:0.669243:0.778315:0.000000
Conclusión: el valor máximo es  ( 1) = 2 y el valor mínimo es  (1) =  2.:@0.293662:0.822748:0.796787:0.822748:0.796787:0.806315:0.293662:0.806315:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f:@0.521286:0.822748:0.525980:0.822748:0.525980:0.806342:0.521286:0.806342:0.000000
–:@0.530710:0.823331:0.539914:0.823331:0.539914:0.805677:0.530710:0.805677:0.000000
f:@0.733650:0.822748:0.738344:0.822748:0.738344:0.806342:0.733650:0.806342:0.000000
–:@0.775564:0.823331:0.784767:0.823331:0.784767:0.805677:0.775564:0.805677:0.000000
Lo descrito en la gráfica del Ejemplo 2 se cumple para las funciones derivables, tal como :@0.293662:0.850110:0.931486:0.850110:0.931486:0.833676:0.293662:0.833676:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
lo indica el siguiente teorema.:@0.293662:0.866752:0.510956:0.866752:0.510956:0.850318:0.293662:0.850318:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
2.2 Teorema de Fermat:@0.293662:0.901256:0.465816:0.901256:0.465816:0.884587:0.293662:0.884587:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Si   tiene un máximo o un mínimo relativo en   y además  ´( ) existe, entonces  ´( ) = 0.     :@0.293662:0.928618:0.927869:0.928618:0.927869:0.912184:0.293662:0.912184:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f:@0.309842:0.928618:0.314536:0.928618:0.314536:0.912212:0.309842:0.912212:0.000000
c:@0.624035:0.928618:0.631693:0.928618:0.631693:0.912212:0.624035:0.912212:0.000000
f c:@0.708598:0.928618:0.731110:0.928618:0.731110:0.912212:0.708598:0.912212:0.000000:0.000000:0.000000
f c:@0.859574:0.928618:0.882086:0.928618:0.882086:0.912212:0.859574:0.912212:0.000000:0.000000:0.000000
Competencia :@0.078581:0.455864:0.187141:0.455864:0.187141:0.438917:0.078581:0.438917:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
matemática :@0.078581:0.472506:0.176057:0.472506:0.176057:0.455559:0.078581:0.455559:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Sea la función f definida :@0.086726:0.492953:0.247867:0.492953:0.247867:0.478013:0.086726:0.478013:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
en un intervalo  ;  :@0.086726:0.508082:0.210691:0.508082:0.210691:0.493142:0.086726:0.493142:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
D:@0.190192:0.508082:0.200667:0.508082:0.200667:0.493167:0.190192:0.493167:0.000000
c:@0.086726:0.523210:0.093688:0.523210:0.093688:0.508296:0.086726:0.508296:0.000000
 :@0.093621:0.523210:0.097285:0.523210:0.097285:0.508271:0.093621:0.508271:0.000000
 :@0.108165:0.522706:0.112349:0.522706:0.112349:0.510237:0.108165:0.510237:0.000000
D:@0.112265:0.523210:0.122741:0.523210:0.122741:0.508296:0.112265:0.508296:0.000000
. Si  ( ) es un valor :@0.122674:0.523210:0.245283:0.523210:0.245283:0.508271:0.122674:0.508271:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f  c:@0.143809:0.523210:0.162299:0.523210:0.162299:0.508296:0.143809:0.508296:0.000000:0.000000:0.000000:0.000000
extremo, entonces :@0.086729:0.538339:0.213580:0.538339:0.213580:0.523400:0.086729:0.523400:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
c debe ser un punto :@0.086729:0.553468:0.224093:0.553468:0.224093:0.538528:0.086729:0.538528:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
crítico. Los puntos :@0.086729:0.568597:0.209815:0.568597:0.209815:0.553657:0.086729:0.553657:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
críticos son de tres :@0.086729:0.583726:0.213006:0.583726:0.213006:0.568786:0.086729:0.568786:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
clases: punto fronterizo :@0.086729:0.598855:0.243594:0.598855:0.243594:0.583915:0.086729:0.583915:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
de  , punto estacionario :@0.086729:0.613984:0.252500:0.613984:0.252500:0.599044:0.086729:0.599044:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
D:@0.107396:0.613984:0.117871:0.613984:0.117871:0.599069:0.107396:0.599069:0.000000
de en donde  ´( ) = 0 :@0.086729:0.629113:0.238521:0.629113:0.238521:0.614173:0.086729:0.614173:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
 f :@0.103798:0.629113:0.113955:0.629113:0.113955:0.614198:0.103798:0.614198:0.000000:0.000000:0.000000
f  c:@0.181861:0.629113:0.205289:0.629113:0.205289:0.614198:0.181861:0.614198:0.000000:0.000000:0.000000:0.000000
o punto singular de f en :@0.086729:0.644242:0.248606:0.644242:0.248606:0.629302:0.086729:0.629302:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
donde  ´( ) no existe.:@0.086729:0.659371:0.227364:0.659371:0.227364:0.644431:0.086729:0.644431:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f  c:@0.134153:0.659371:0.157580:0.659371:0.157580:0.644456:0.134153:0.644456:0.000000:0.000000:0.000000:0.000000
MUESTRA EDITORIAL:@0.248902:0.947499:0.888171:0.113285:0.743922:0.050540:0.104653:0.884754:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
 Muestra editorial solo para fines didácticos – Prohibida su venta:@0.095585:0.659700:0.095585:0.341447:0.077558:0.341447:0.077558:0.659700:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000