Find the local maximum and minimum of $LaTeX: \displaystyle f(x) = 7 x^{3} + 17 x^{2} + 9 x - 2$ .

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = 21 x^{2} + 34 x + 9$ . Solving $LaTeX: \displaystyle 21 x^{2} + 34 x + 9 = 0$ gives $LaTeX: \displaystyle x = \left[ - \frac{9}{7}, \ - \frac{1}{3}\right]$ . Using the 2nd derivative test gives:
$LaTeX: \displaystyle f''\left( - \frac{9}{7} \right) = -20$ which is less than zero, so the function is concave down and $LaTeX: \displaystyle f\left(- \frac{9}{7}\right) = - \frac{17}{49}$ is a local maximum.
$LaTeX: \displaystyle f''\left( - \frac{1}{3} \right) = 20$ which is greater than zero, so the function is concave up and $LaTeX: \displaystyle f\left(- \frac{1}{3}\right) = - \frac{91}{27}$ is a local minimum.