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Problem 37

a. Find the open intervals on which the function is increasing and decreasing.

b. Identify the function's local and absolute extreme values, if any, saying where they occur.

$$

f(x)=x^{1 / 3}(x+8)

$$

Answer

\documentclass{article}

\usepackage{amsmath , amssymb ,amsthm}

\usepackage[utf8]{inputenc}

\begin{document}

a) To find out intervals on which the function is increasing or decreasing we need to find out the sign of the derivative function. Because the derivative of a function shows its slop in each point. When the slop is positive the function is increasing and when the slop is negative the function is decreasing.\\

We have

$$f(x) = x^{4/3}+8x^{1/3} $$

And rthe derivative of this function is

$$f'(x) = \frac{4}{3}x^{1/3}+\frac{8]{3}x^{-2/3} $$

The derivative function is always positive and to make sure about this result we find $f'(x)=0$ points, which are -2 and 0. -2 is not acceptable because it cauese a root of negative number. The only root is 0 and for $x>0$, $f'(x)$ is positive. \\

The domain of the original function is $[0,\inf)$ so the function is positive in all its domain. \\

b) As the only root of $f'(x)$ is 0 and $f'(0)=\inf$ then 0 is saddle point and the function does not have any minimum or maximum.

\end{document}

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Graph

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Graph

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