How do you find the x and y coordinates of all inflection points #f(x) = x^4 - 12x^2#?

1 Answer
Sep 3, 2015

The coordinates of the two inflection points are #(x,y)=(pm sqrt(2),-20)#

Explanation:

The first derivative is #f'(x)=4x^3-24x# and the second derivative is #f''(x)=12x^2-24=12(x^2-2)#.

The second derivative is zero only at #x=pm sqrt(2)# and, in fact, changes sign as #x# increases through these two values. Therefore the #x#-coordinates of the two inflection points are #x=pm sqrt(2)#.

Since #f(pm sqrt(2))=4-12*2=4-24=-20#, it follows that the coordinates of the two inflection points are #(x,y)=(pm sqrt(2),-20)#.

Here's the graph. See if you can find the inflection points in the graph:

graph{x^4-12x^2 [-10, 10, -40,40]}