Question

Draw the graph of ${\sin }^{2}x$ and $|\sin x|$ and show the continuity and differentiability of both function.

Answer 1

Consider the diagram shown above.

This diagram shows the simultaneous graphs of ${\sin }^{2}x$ and $|\sin x|$ on same cartesian plane.

Here, we can see that both the functions are continuous everywhere.

But, while ${\sin }^{2}x$ doesn't have any sharp edges, $|\sin x|$ has many sharp edges (sharp turning points). We know that, if any function takes sharp turns at any point, it cannot be differentiable at that point. Therefore, ${\sin }^{2}x$ is continuous and differntiable at everywhere, the function $|\sin x|$ is continuous everywhere but not differentiable at points where $\sin x=0$.