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

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = - 3 x^{2} + 8 x - 5$ . Solving $LaTeX: \displaystyle - 3 x^{2} + 8 x - 5 = 0$ gives $LaTeX: \displaystyle x = \left[ 1, \ \frac{5}{3}\right]$ . Using the 2nd derivative test gives:
$LaTeX: \displaystyle f''\left( 1 \right) = 2$ which is greater than zero, so the function is concave up and $LaTeX: \displaystyle f\left(1\right) = 1$ is a local minimum.
$LaTeX: \displaystyle f''\left( \frac{5}{3} \right) = -2$ which is less than zero, so the function is concave down and $LaTeX: \displaystyle f\left(\frac{5}{3}\right) = \frac{31}{27}$ is a local maximum.