Question

Use the definitions of increasing and decreasing to argue that $f\left(x\right)=x^4$ is decreasing on $(-\infty ,0]$ and increasing on $[0,\infty )$. Then use derivatives to argue the same thing.

Short answer

The statement has been proven.

Step-by-step solution

Step 1. Given information

We have been given a function $f\left(x\right)=x^4$.

We have to use the definitions of increasing and decreasing to argue that this function is decreasing on $(-\infty ,0]$and increasing on $[0,\infty )$.

Step 2. Using the definition

For a and b in the interval $(-\infty ,0]$

Now if $a<b\le 0$

Then,

role="math" localid="1648442267515" $a^4>b^4$

Thus the function is decreasing in the interval $(-\infty ,0]$

Also,

For a and b in the interval role="math" localid="1648442301380" $[0,\infty )$

Now if $0<a<b$then,

$a^4<b^4$

Thus the function is increasing in the interval$[0,\infty )$

Step 3. Using the derivative

The derivative of the function is given by :

$f^{\mathrm{\prime }}\left(x\right)=4x^3$

The function $f^{\mathrm{\prime }}\left(x\right)=4x^3$is always negative for x<0

The function $f^{\mathrm{\prime }}\left(x\right)=4x^3$is always positive for x>0