Aké sú extrémy h (x) = 7x ^ 5 - 12x ^ 3 + x?

odpoveď:

Extrémia je na x =#+-1# a x =# + - sqrt (1/35) #

h (x) = # 7x ^ 5 -12x ^ 3 + x #

h '(x) = # 35x ^ 4 -36x ^ 2 + 1 #

Faktorizácia h '(x) a jej vyrovnanie na nulu by to bolo# (35x ^ 2 -1) (x ^ 2-1) = 0 #

Kritické body sú preto # + - 1, + -sqrt (1/35) #

h '' (x) = # 140x ^ 3-72x #

Pre x = -1, h '' (x) = -68, teda by boli maximá v x = -1

pre x = 1, h '' (x) = 68, teda by boli minimá v x = 1

pre x =#sqrt (1/35) #, h '' (x) = 0,6761- 12,1702 = - 11,4941, preto by v tomto bode existovala maxima

pre x = # -sqrt (1/35), h '' (x) = -0,6761 + 12,1702 = 11,4941, preto by v tomto bode boli minimá.