Find the local maximum and minimum of $LaTeX: \displaystyle f(x) = - x^{3} + 13 x^{2} + 9 x - 4$ .

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = - 3 x^{2} + 26 x + 9$ . Solving $LaTeX: \displaystyle - 3 x^{2} + 26 x + 9 = 0$ gives $LaTeX: \displaystyle x = \left[ - \frac{1}{3}, \ 9\right]$ . Using the 2nd derivative test gives:
$LaTeX: \displaystyle f''\left( - \frac{1}{3} \right) = 28$ which is greater than zero, so the function is concave up and $LaTeX: \displaystyle f\left(- \frac{1}{3}\right) = - \frac{149}{27}$ is a local minimum.
$LaTeX: \displaystyle f''\left( 9 \right) = -28$ which is less than zero, so the function is concave down and $LaTeX: \displaystyle f\left(9\right) = 401$ is a local maximum.