Mathematics - Continuity - Differentiability Question with Solution | TestHub

$f(x)=\left(x^2+1\right)\left|x^2-ax+2\right|+\cos |x|$ Notice that $\cos (-x)=\cos x=\cos |x|$ which means $\cos |x|$ is differentiable everywhere in $x\in R$ $\begin{array}{rl}& \Rightarrow f(x)\text{ can be non differentiable where }\left|x^2-ax+2\right|\\ & =0\end{array}$ $\Rightarrow x^2-ax+2=0$
$\Rightarrow 4-2a+2=0\Rightarrow a=3$ $\Rightarrow \left(x^2-3x+2\right)=0\Rightarrow x=1,2$ $\beta =1$ distance of $(\alpha ,\beta )$ from line $\begin{array}{rl}& 12x+5y+10=0\\ & \Rightarrow \frac{|2(12)+5(1)+10|}{13}=\frac{39}{13}=3\end{array}$