Question

If the function $f\left(x\right)$ is continuous in the interval $[-2,2]$, find the values of $a$ and $b$ where $f\left(x\right)=\{\begin{array}{c}\frac{\sin ax}{x}-2,for-2\le x\le 0\\ 2x+1,for0\le x\le 1\\ 2b\sqrt{x^2+3}-1,for1\le x\le 2\end{array}$

Answer 1

$f\left(1\right)=2\left(1\right)+1=3$

$\Rightarrow f\left(1^{-}\right)=3$

$f\left(1^{+}\right)=2b\sqrt{4}-1=4b-1=3$

$b=1$

$\Rightarrow f\left(0\right)=f\left(0^{-}\right)=f\left(o^{+}\right)$

$a-2=1$

$a=3$

Hence, the answer is $3,1.$