Càlcul de màxims i mínims d'una funció

MAX I MIN

f(x)=4x^4-4x^3-2x^2+12x

1.Fer derivada de la funció inicial -->f'(x)=4x^3-12x^2-4x+12 --> rufini

2.Cercar punts q anulen la derivada (rufini/b+-...)

rufini =4(x-3).(x^2-1)=0         4(x-3)->x=3       (x^2-1)->x^2-1=0  x^2=1  X=+-raiz 1=+-1  

3.Fer la segona derivada per veure si son max o min

F''(x)=12x^2-24x-4

4.Substituir punts singulars(3/+-1)a la segona derivada

5.Mirar si   x<0 max          x>0 min

f''(3) =12(3)^2-24(3)-4= 40 >0 ->min                                                                                                                                                                                                       F''(1) = 12(1)^2-24(1)-4= -16<0 ->max                                                                                                                                                                                                                  F''(-1)=12(-1)^2-24(-1)-4=32 >0->min

6.Substitiur punts singulars a la funció inicial

f(3)=  3^4-4(3)^3-2(3)^2+12(3)= -9    punt(3,-9)-->min                                                                                                                                                                                                                            f (1)=  1^4-4(1)^3-2(1)^2+12(1)= 7    Punt(1,7)--> max                                                                                                                                                                                                                        f(-1)=(-1)^4-4(-1)^3-2(-1)^2+12(-1)= -9 punt(-1,-9)-->min

DERIVADA DEFINICIÓ

lim h->0   F(a+h)-f(a)                                                                                                                                                                                        _________________________________  =f'(a)                                                                                                                                                                                                     h

1.Aplicar i substituir si cal

2. Calcular les dos "parts" de la formula(  f(a+h)    F(a)  )

3.Substituir resultat de les "parts" a la formula inicial

4.Arreglar o simplificar

5.Treure factor comú

6.Substituir la "h".Per comprovar fer derivada "facil"