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

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = - 9 x^{2} - 2 x + 7$ . Solving $LaTeX: \displaystyle - 9 x^{2} - 2 x + 7 = 0$ gives $LaTeX: \displaystyle x = \left[ -1, \ \frac{7}{9}\right]$ . Using the 2nd derivative test gives:
$LaTeX: \displaystyle f''\left( -1 \right) = 16$ which is greater than zero, so the function is concave up and $LaTeX: \displaystyle f\left(-1\right) = -8$ is a local minimum.
$LaTeX: \displaystyle f''\left( \frac{7}{9} \right) = -16$ which is less than zero, so the function is concave down and $LaTeX: \displaystyle f\left(\frac{7}{9}\right) = \frac{104}{243}$ is a local maximum.