### Home > APCALC > Chapter 5 > Lesson 5.1.2 > Problem5-16

5-16.

Without your graphing calculator, sketch $y = f\left(x\right)$ and algebraically determine where the function is increasing, decreasing, concave up, and concave down. List any maxima, minima, and points of inflection.

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

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

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

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