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Q. 11

Expert-verifiedFound in: Page 260

Book edition
1st

Author(s)
Peter Kohn, Laura Taalman

Pages
1155 pages

ISBN
9781429241861

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.

The statement has been proven.

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 $[0,\infty )$.

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 $[0,\infty )$

The derivative of the function is given by :

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

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

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

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