Find the local maximum and minimum of $LaTeX: \displaystyle f(x) = - 3 x^{3} - 9 x^{2} + 7 x + 3$ .

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = - 9 x^{2} - 18 x + 7$ . Solving $LaTeX: \displaystyle - 9 x^{2} - 18 x + 7 = 0$ gives $LaTeX: \displaystyle x = \left[ - \frac{7}{3}, \ \frac{1}{3}\right]$ . Using the 2nd derivative test gives:
$LaTeX: \displaystyle f''\left( - \frac{7}{3} \right) = 24$ which is greater than zero, so the function is concave up and $LaTeX: \displaystyle f\left(- \frac{7}{3}\right) = - \frac{218}{9}$ is a local minimum.
$LaTeX: \displaystyle f''\left( \frac{1}{3} \right) = -24$ which is less than zero, so the function is concave down and $LaTeX: \displaystyle f\left(\frac{1}{3}\right) = \frac{38}{9}$ is a local maximum.