3. Consider the function, $f(x) = x^3 - x^2$. Answer each of the following:
(i) Verify that, $f(x)$ satisfies the conditions of Mean Value Theorem in the interval $[-1, 1]$ and find all values of $c$ such that $f'(c) = \frac{f(1) - f(-1)}{1 - (-1)}$.
(ii) Identify the first and second order critical points of $f(x)$; identify the intervals where the function $f(x)$ is increasing, where the function $f(x)$ is decreasing; and the graph of $f(x)$ is concave up and where the graph is concave down; classify the first order critical points as local maximum or local minimum, if any, and classify the second order critical points, if any as inflection points or not. And then sketch the graph of $f(x)$ reflecting your above findings.
(iii) Find the absolute maximum and absolute minimum of $f(x)$ in $[-1, 1]$.
