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Problem 38

a. Find the open intervals on which the function …


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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)


\usepackage{amsmath , amssymb ,amsthm}
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.



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