Find the local maximum and minimum of $LaTeX: \displaystyle f(x) = - 3 x^{3} + 12 x^{2} + 9 x + 1$ .

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = - 9 x^{2} + 24 x + 9$ . Solving $LaTeX: \displaystyle - 9 x^{2} + 24 x + 9 = 0$ gives $LaTeX: \displaystyle x = \left[ - \frac{1}{3}, \ 3\right]$ . Using the 2nd derivative test gives:
$LaTeX: \displaystyle f''\left( - \frac{1}{3} \right) = 30$ which is greater than zero, so the function is concave up and $LaTeX: \displaystyle f\left(- \frac{1}{3}\right) = - \frac{5}{9}$ is a local minimum.
$LaTeX: \displaystyle f''\left( 3 \right) = -30$ which is less than zero, so the function is concave down and $LaTeX: \displaystyle f\left(3\right) = 55$ is a local maximum.