Question

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

From graph, we can say that both the functions $y=\sin x\&y={\sin }^{-1}(\cos x)$ 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.