Find the local maximum and minimum of $LaTeX: \displaystyle f(x) = 9 x^{3} - 39 x^{2} - 9 x + 2$ .

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = 27 x^{2} - 78 x - 9$ . Solving $LaTeX: \displaystyle 27 x^{2} - 78 x - 9 = 0$ gives $LaTeX: \displaystyle x = \left[ - \frac{1}{9}, \ 3\right]$ . Using the 2nd derivative test gives:
$LaTeX: \displaystyle f''\left( - \frac{1}{9} \right) = -84$ which is less than zero, so the function is concave down and $LaTeX: \displaystyle f\left(- \frac{1}{9}\right) = \frac{203}{81}$ is a local maximum.
$LaTeX: \displaystyle f''\left( 3 \right) = 84$ which is greater than zero, so the function is concave up and $LaTeX: \displaystyle f\left(3\right) = -133$ is a local minimum.