extrema and mean value theorem

[sully405]

Find the absolute extrema of the following functions on the given intervals.
a.) f(x)=(x^(2/3))(x^2-1) [-1,1]

b.) f(x)=(x^(1/2))/(x^2 +1) [1/4,1/2]

Find all values of c that satisfy the conclusion of the Mean Value Theorem of the following function:

f(x)=x^(1/3)-x [-8,-1][/code]

 

[pka]

First find the derivative: \(\displaystyle f(x) = \sqrt[3]{x} - x,\quad f'(x) = \frac{1}{{\sqrt[3]{{x^2 }}}} - 1\)

Now you need to solve for c: \(\displaystyle f'(c) = \frac{{f( - 1) - f( - 8)}}{{\left( { - 1} \right) - \left( { - 8} \right)}} = \frac{{0 - \left( 6 \right)}}{7}\)