Find the local maximum and minimum of $LaTeX: \displaystyle f(x) = 5 x^{3} - 11 x^{2} - 9 x + 4$ .

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = 15 x^{2} - 22 x - 9$ . Solving $LaTeX: \displaystyle 15 x^{2} - 22 x - 9 = 0$ gives $LaTeX: \displaystyle x = \left[ - \frac{1}{3}, \ \frac{9}{5}\right]$ . Using the 2nd derivative test gives:
$LaTeX: \displaystyle f''\left( - \frac{1}{3} \right) = -32$ which is less than zero, so the function is concave down and $LaTeX: \displaystyle f\left(- \frac{1}{3}\right) = \frac{151}{27}$ is a local maximum.
$LaTeX: \displaystyle f''\left( \frac{9}{5} \right) = 32$ which is greater than zero, so the function is concave up and $LaTeX: \displaystyle f\left(\frac{9}{5}\right) = - \frac{467}{25}$ is a local minimum.