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

Answer:

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

Step-by-step 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.

$\displaystyle\frac{d(f(x))}{dx} = 4x^3 - 16x$

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.