Find the local maximum and minimum of $LaTeX: \displaystyle f(x) = 9 x^{3} + 12 x^{2} + 5 x + 1$ .

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = 27 x^{2} + 24 x + 5$ . Solving $LaTeX: \displaystyle 27 x^{2} + 24 x + 5 = 0$ gives $LaTeX: \displaystyle x = \left[ - \frac{5}{9}, \ - \frac{1}{3}\right]$ . Using the 2nd derivative test gives:
$LaTeX: \displaystyle f''\left( - \frac{5}{9} \right) = -6$ which is less than zero, so the function is concave down and $LaTeX: \displaystyle f\left(- \frac{5}{9}\right) = \frac{31}{81}$ is a local maximum.
$LaTeX: \displaystyle f''\left( - \frac{1}{3} \right) = 6$ which is greater than zero, so the function is concave up and $LaTeX: \displaystyle f\left(- \frac{1}{3}\right) = \frac{1}{3}$ is a local minimum.