Question

If the function $f\left(x\right)=\{\begin{array}{cc}x+2& \text{if}x<2\\ a{x}^2+bx+3& \text{if}2\le x<3\\ 2x+a+b& \text{if}x\ge 3\end{array}$ is continous, then the value of $(a^2+b^2)$ is

• A. $\frac{1}{2}$
• B. $\frac{1}{3}$
• C. $\frac{1}{6}$
• D. $\frac{1}{9}$

Answer 1

The correct option is A $\frac{1}{2}$

$f\left(x\right)$ is continous at $x=2$
$\Rightarrow 4=4a+2b+3\cdots \left(1\right)$
$f\left(x\right)$ is continous at $x=3$
$\Rightarrow 9a+3b+3=6+a+b\cdots \left(2\right)\Rightarrow a=\frac{1}{2},b=-\frac{1}{2}\Rightarrow a^2+b^2=\frac{1}{2}$