CPM Homework Help : CALC Problem 5-16

Without your graphing calculator, sketch $f(x)$ and algebraically determine where the function is increasing, decreasing, concave up, and concave down. List any maximums, minimums, and points of inflection.

$f ( x ) = \left\{ \begin{array} { c c } { - ( x - 2 ) ^ { 2 } + 3 } & \text{for } { x \leq 2 } \\ { ( x - 3 ) ^ { 2 } + 2 } & { \text { for } x > 2 } \end{array} \right.$

Hint 1:

A function is increasing where its first derivative $> 0$ and decreasing where its first derivative is $< 0$ .
Find these intervals algebraically.

Hint 2:

A function is concave up when its second derivative is $> 0$ and concave down when its second derivative is $< 0$ .
Find these intervals algebraically.

Hint 3:

Notice that $x = 2$ is a boundary point. Even though $f^\prime(2) ≠ 0$ and $f^{\prime\prime}(2) ≠ 0$ , interesting changes in slope and concavity can still happen at $x = 2$ .