Question

Let $f\left(x\right)$ be defined in $[–2,2]$ by $f\left(x\right)=\{\begin{array}{cc}max\{\sqrt{4-x^2},\sqrt{1+x^2}\}& -2\le x\le 0\\ min\{\sqrt{4-x^2},\sqrt{1+x^2}\}& 0<x\le 2\end{array}$ . Then $f\left(x\right)$ is

• A. Continuous & differentiable at all points
• B. Discontinuous at one point & non-differentiable at more than one point
• C. Non-differentiable at more than one point & continuous at all points
• D. Discontinuous & non-differentiable at only one point

Answer 1

The correct option is B

Discontinuous at one point & non-differentiable at more than one point

$y=\sqrt{4-x^2}$

$\Rightarrow y^2=4-x^2$

$\Rightarrow x^2+y^2=4$ ... (1)

It is a circle with centre at $(0,0)$ and radius equal to 2

and $y=\sqrt{1+x^2}$

$\Rightarrow y^2=1+x^2$

$\Rightarrow y^2-x^2=1$ ... (2) which is a hyperbola

Then $f\left(x\right)$ is