How do you find points of inflection of the function function #h(x) = −x^4 + x^2 − 1#?

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Answer 1 · 2015-04-21T22:34:18

The points of inflection are #x_1=-sqrt(6)/6# #x_2=sqrt(6)/6#

To calculate points of inflection you have to find zeros of the 2nd derivative and check if it changes sign at those points.

#h'(x)=-4x^3+2x#
#h''(x)=-12x^2+2#

Looking for the points where #h''(x)=0#

#-12x^2+2=0 ////:2#
#-6x^2+1=0#
#6x^2=1#
#x^2=1/6#
#x_1=-sqrt(6)/6# or #x_2=sqrt(6)/6#

Checking if the derivative changes sign

graph{-12x^2+2 [-4.506, 4.505, -2.252, 2.254]}

From the graph we see that the derivative changes sign in both points #x_1# and #x_2# so they are the points of inflection.