Find the local maximum and minimum of $LaTeX: \displaystyle f(x) = 3 x^{3} + 12 x^{2} + 7 x - 4$ .

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = 9 x^{2} + 24 x + 7$ . Solving $LaTeX: \displaystyle 9 x^{2} + 24 x + 7 = 0$ gives $LaTeX: \displaystyle x = \left[ - \frac{7}{3}, \ - \frac{1}{3}\right]$ . Using the 2nd derivative test gives:
$LaTeX: \displaystyle f''\left( - \frac{7}{3} \right) = -18$ which is less than zero, so the function is concave down and $LaTeX: \displaystyle f\left(- \frac{7}{3}\right) = \frac{62}{9}$ is a local maximum.
$LaTeX: \displaystyle f''\left( - \frac{1}{3} \right) = 18$ which is greater than zero, so the function is concave up and $LaTeX: \displaystyle f\left(- \frac{1}{3}\right) = - \frac{46}{9}$ is a local minimum.