Question

At what value(s) of x does f(x)=x^4-18x^2 have a critical point where the graph changes from decreasing to increasing?

Answer 1

take f"(x)=0 to find critical values v1,v2,...
take f (vn)
produces inflection points

Answer 2

`x = 2, x = -2` are the critical points where the graph changes from decreasing to increasing.

Explanation:

• Critical points are the points on the where the function either a changes from increasing to decreasing or from decreasing to increasing.
• Critical points are useful for determining maxima, minima and solving optimization problems.

We are given the expression:

`f(x) = x^4 - 18x^2`

We have to find the critical points of the given expression.

First we differentiate the given expression with respect to x.

`\displaystyle(d(f(x)))/(dx) = 4x^3 - 16x`

Equating f'(x) to zero, to obtain the critical points:

`f'(x) = 0\\4x^3 - 16x = 0\\x(4x^2 - 16) = 0\\x(2x-4)(2x+4) = 0\\x = 0, x = 2, x = -2`

Hence,
`x = 2, x = -2` are the critical points where the graph changes from decreasing to increasing.