CPM Homework Help : APCALC Problem 7-95

Determine values for $a, b, c$ , and $d$ such that the function given at right is continuous and differentiable for all values of $x$ .

$f ( x ) = \left\{ \begin{array} { c c c } { 2 ^ { x } } & { \text { for } } & { x < 0 } \\ { a x ^ { 2 } + b x + c } & { \text { for } } & { 0 \leq x \leq 4 } \\ { d } & { \text { for } } & { x > 4 } \end{array} \right.$

Hint:

Continuous implies that the pieces of the function agree at the boundary points.
Differentiable implies that the derivatives of the pieces agree at the boundary points.

Steps:

For each boundary point, $x = 0$ and $x = 4$ , write and solve a system of equations.
One equation connects $f(x)$ , the other connects $f^\prime(x)$ .