Find the local maximum and minimum of $LaTeX: \displaystyle f(x) = 9 x^{3} + 12 x^{2} - 3 x + 1$ .

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = 27 x^{2} + 24 x - 3$ . Solving $LaTeX: \displaystyle 27 x^{2} + 24 x - 3 = 0$ gives $LaTeX: \displaystyle x = \left[ -1, \ \frac{1}{9}\right]$ . Using the 2nd derivative test gives:
$LaTeX: \displaystyle f''\left( -1 \right) = -30$ which is less than zero, so the function is concave down and $LaTeX: \displaystyle f\left(-1\right) = 7$ is a local maximum.
$LaTeX: \displaystyle f''\left( \frac{1}{9} \right) = 30$ which is greater than zero, so the function is concave up and $LaTeX: \displaystyle f\left(\frac{1}{9}\right) = \frac{67}{81}$ is a local minimum.