Find the local maximum and minimum of $LaTeX: \displaystyle f(x) = - 9 x^{3} + 6 x^{2} + 7 x - 1$ .

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = - 27 x^{2} + 12 x + 7$ . Solving $LaTeX: \displaystyle - 27 x^{2} + 12 x + 7 = 0$ gives $LaTeX: \displaystyle x = \left[ - \frac{1}{3}, \ \frac{7}{9}\right]$ . 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 $LaTeX: \displaystyle f\left(- \frac{1}{3}\right) = - \frac{7}{3}$ is a local minimum.
$LaTeX: \displaystyle f''\left( \frac{7}{9} \right) = -30$ which is less than zero, so the function is concave down and $LaTeX: \displaystyle f\left(\frac{7}{9}\right) = \frac{311}{81}$ is a local maximum.