Find the local maximum and minimum of $LaTeX: \displaystyle f(x) = - 7 x^{3} - 6 x^{2} + 9 x - 3$ .

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = - 21 x^{2} - 12 x + 9$ . Solving $LaTeX: \displaystyle - 21 x^{2} - 12 x + 9 = 0$ gives $LaTeX: \displaystyle x = \left[ -1, \ \frac{3}{7}\right]$ . Using the 2nd derivative test gives:
$LaTeX: \displaystyle f''\left( -1 \right) = 30$ which is greater than zero, so the function is concave up and $LaTeX: \displaystyle f\left(-1\right) = -11$ is a local minimum.
$LaTeX: \displaystyle f''\left( \frac{3}{7} \right) = -30$ which is less than zero, so the function is concave down and $LaTeX: \displaystyle f\left(\frac{3}{7}\right) = - \frac{39}{49}$ is a local maximum.