Question

If $f\left(x\right)=max\left\{sin x , \left(sin\right)^{- 1} \left(cos x\right)\right\}$ , then

• A. $f$ is differentiable everywhere
• B. $f$ is continues everywhere but not differentiable
• C. $f$ is discontinuous at $x=\frac{n \pi }{2},n\in I$
• D. $f$ is non $-$ differentiable at $x=\frac{n \pi }{2},n\in I$

Answer 1

Answer: (B)

Given $f\left(x\right)=max\left\{sin x , \left(sin\right)^{- 1} \left(cos x\right)\right\}$ .
Sketching the graph of both the functions $y=sinx\&y=\left(sin\right)^{- 1}\left(cos x\right)$ on the same co-ordinate plane:

From graph, we can say that both the functions $y=\sin x \quad \& \quad y=\sin ^{-1}(\cos x)$ are continuous for all real values of $x$.
Also, $y=\sin ^{-1}(\cos x)$ has infinite corner points. So, it is non-differentiable at those corner points and hence when $f(x)=\sin ^{-1}(\cos x)$, it is nondifferentiable at infinite points.
So, clearly $f(x)$ is continuous for all real values of $x$ but non-differentiable at infinite points.