Calclus question help please! "Use the derivative to determine whether the function is strictly monotonic

Question

Use the derivative to determine whether the function is strictly monotonic on its entire domain and therefore has an inverse function

f(x)=(x^4/4)-2x^2

so first i take a derivative and then how then do i graph it and see whether it is constantly increasing or decreasing?

how do i do this problem? im lost

Help please

i woudl really appreciate it

Answer 1

f(x) =( x^4/4) - 2x^2

df/dx = x^3 - 4x = x(x^2 - 4) = 0

The solutions of this are: x = 0. x = 2 and x = -2

There are four regions then that need to be examined to see how the derivative behaves in each of them:

-Inf to -2

-2 to 0

0 to 2

2 to +Inf

A monotonic function can have the derivative equal 0 at points and that is OK. For an increasing function the derivative in each of these intervals should be positive. For a decreasing function the derivative in each of these interals should be negative.

Test each of these regions using a selected value of x:

-Inf to -2:  Use x = -10 and get df/dx < 0

So the function is decreasing over this range.

-2 to 0: Use x = -1 and get df/dx > 0

So the function is increasing over this range

A monotonic function is one whose successive values are either increasing or decreasing. Therefore this function is not strictly monotonic since it both increases and decreases.