CPM Homework Help : APCALC Problem 12-84

Multiple Choice: Which of the following statements must be true about the function defined at right?

$f ( x ) = \left\{ \begin{array} { l l } { e ^ { x + 2 } - 1 } & { \text { for } \quad x \leq - 2 } \\ { 4 - x ^ { 2 } } & { \text { for } - 2 < x < 0.5 } \\ { 4 - \operatorname { cos } ( x - 0.5 ) } & { \text { for } \quad x \geq 0.5 } \end{array} \right.$

$f$ is continuous at $x = -2$ .
$f$ is differentiable at $x = -2$ .
$f$ is differentiable at $x = 0.5$ .

I only

II only

I and III only

II and III only

I and II only

Steps I:

$e^{-2+2}-1=0\text{ and }4-(-2)^2=0$

Therefore $f$ is continuous at $x = -2$ .

Steps II:

$e^{-2+2}=1\text{ and }4-2(-2)=0$

Therefore $f$ is not differentiable at $x = –2$ .