Question

Draw the graph of $si{n}^{2}x$ and $|sinx|$ and show the continuity and differentiability of both the functions.

Answer 1

Hint: To show the continuity of a function, we should ensure that it exists at all points and there are no breaks or sharp edges on the graph of that function. To check the differentiability of a function $f(x)$ at a point, the formula is-
$\underset{\mathrm{h}\to 0}{lim}{\displaystyle \frac{\mathrm{f}(\mathrm{x}+\mathrm{h})-\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{h}}}\mathrm{exists}\mathrm{at}\mathrm{all}\mathrm{values}\mathrm{of}\mathrm{x}$

Complete step by step answer:
The graphs of the two functions are-

Here the the graph which is inside is $si{n}^{2}x$ and the outer one is $|sinx|$. From the graph it is clearly visible that $si{n}^{2}x$ is smooth all along but $|sinx|$ has a sharp curve when it touches the x-axis.
Since $si{n}^{2}x$ is smooth at all points, it is continuous and differentiable at every point.
Since $|sinx|$ has sharp curves when it touches the x-axis, it is neither continuous nor differentiable at those points.

This is the required answer.

Note: Initially when looking at the graph, it seems that both the functions are perfectly smooth, but it is not right. Due to the presence of modulus function, $|sinx|$ changes direction abruptly. But $si{n}^{2}x$ changes the direction in a smooth manner.