# Find the constants of A and B?

Feb 9, 2017

$\left(a , b\right) = \left(- 1 , 6\right)$

#### Explanation:

If $f \left(x\right)$ is continuous for $- 3 \le x \le 0$
then it is continuous at $x = - 2$ and at $x = 0$

If $f \left(x\right)$ is continuous at $x = - 2$
then
$\textcolor{w h i t e}{\text{XXX}} f \left(x\right) = - x$ at $x = - 2$
$\textcolor{w h i t e}{\text{XXX}} \rightarrow f \left(- 2\right) = a \cdot {\left(- 2\right)}^{2} + b = - \left(- 2\right)$
$\textcolor{w h i t e}{\text{XXX}} \rightarrow 4 a + b = 2$

If $f \left(x\right)$ is continuous at $x = 0$
then
$\textcolor{w h i t e}{\text{XXX}} f \left(x\right) = 6$ at $x = 0$
$\textcolor{w h i t e}{\text{XXX}} \rightarrow f \left(0\right) = a \cdot {0}^{2} + b = 6$
$\textcolor{w h i t e}{\text{XXX}} \rightarrow b = 6$

Combining $4 a + b = 2$ and $b = 6$
gives
$\textcolor{w h i t e}{\text{XXX}} a = - 1$

Graphically here is what we have: