Find the local maximum and minimum of $LaTeX: \displaystyle f(x) = 3 x^{3} + 7 x^{2} + 5 x + 2$ .

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = 9 x^{2} + 14 x + 5$ . Solving $LaTeX: \displaystyle 9 x^{2} + 14 x + 5 = 0$ gives $LaTeX: \displaystyle x = \left[ -1, \ - \frac{5}{9}\right]$ . Using the 2nd derivative test gives:
$LaTeX: \displaystyle f''\left( -1 \right) = -4$ which is less than zero, so the function is concave down and $LaTeX: \displaystyle f\left(-1\right) = 1$ is a local maximum.
$LaTeX: \displaystyle f''\left( - \frac{5}{9} \right) = 4$ which is greater than zero, so the function is concave up and $LaTeX: \displaystyle f\left(- \frac{5}{9}\right) = \frac{211}{243}$ is a local minimum.