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

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = 3 x^{2} + 6 x + 3$ . Solving $LaTeX: \displaystyle 3 x^{2} + 6 x + 3 = 0$ gives $LaTeX: \displaystyle x = \left[ -1\right]$ . Using the 2nd derivative test gives:
$LaTeX: \displaystyle f''\left( -1 \right) = 0$ The 2nd derivative test fails.