Find the local maximum and minimum of $LaTeX: \displaystyle f(x) = 5 x^{3} + 16 x^{2} + 9 x - 4$ .

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = 15 x^{2} + 32 x + 9$ . Solving $LaTeX: \displaystyle 15 x^{2} + 32 x + 9 = 0$ gives $LaTeX: \displaystyle x = \left[ - \frac{9}{5}, \ - \frac{1}{3}\right]$ . Using the 2nd derivative test gives:
$LaTeX: \displaystyle f''\left( - \frac{9}{5} \right) = -22$ which is less than zero, so the function is concave down and $LaTeX: \displaystyle f\left(- \frac{9}{5}\right) = \frac{62}{25}$ is a local maximum.
$LaTeX: \displaystyle f''\left( - \frac{1}{3} \right) = 22$ which is greater than zero, so the function is concave up and $LaTeX: \displaystyle f\left(- \frac{1}{3}\right) = - \frac{146}{27}$ is a local minimum.