Examining differentiability Using Graph of Function - Practice Questions & MCQ

Examining Differentiability Using Differentiation and Graph

1. Using Differentiation (only for continuous functions)

at the split point.

$f(x)=\{\begin{array}{ll}{g}_1(x),& x<a\\ g_2(x),& x\ge a\end{array}$
First, check if $f(x)$ is continuous at $x=a$. If it is not continuous, then it cannot be differentiable.
If it is continuous, then to check differentiability, find

$f^{\prime }(x)=\{\begin{array}{ll}{(g_1(x))}^{\prime },& x<a\\ {(g_2(x))}^{\prime },& x>a\end{array}$
Differentiability can be checked at $\mathrm{x}=\mathrm{a}$ by comparing

$\underset{x\to a^{-}}{lim}{(g_1(x))}^{\prime }\text{ and }\underset{x\to a^{+}}{lim}{(g_2(x))}^{\prime }$

2. Differentiability using Graphs

A function $f(x)$ is not differentiable at $x=a$ if
1. Function is discontinuous at $x=a$
2. The graph of a function has a sharp turn at $x=a$
3. Function has a vertical tangent at $\mathrm{x}=\mathrm{a}$

Illustration 1

Check the differentiability of the following function.
1. $f(x)=\sin |x|$

Method 1
Using graphical transformation, we can draw its graph

Using the graph we can tell that at $x=0$, the graph has a sharp turn, so it is not differentiable at $x=0$.

Method 2

As LHL $=$ RHL $=f(0)=0$, so the function is continuous at $x=0$
So we can use differentiation to check differentiability

$\begin{array}{rl}& \mathrm{f}(\mathrm{x})=\{\begin{array}{cl}-\sin x,& x<0\\ \sin x,& x\ge 0\end{array}\\ & \therefore \\ & \therefore {\mathrm{f}}^{\prime }(\mathrm{x})=\{\begin{array}{cc}-\cos x,& x<0\\ \cos x,& x>0\end{array}\end{array}$
$\therefore \mathrm{LHD}={\mathrm{f}}^{\prime }\left(0^{-}\right)=-1\text{ and }\mathrm{RHD}={\mathrm{f}}^{\prime }\left(0^{+}\right)=1$
As these are not equal, so, $f(x)=\sin |x|$ is not differentiable at $x=0$

Illustration 2

$f(x)=\Vert \log |x|\Vert ,x\text{ not equal to }0$
Plot the graph of | log $|\mathrm{x}|\mid$ using graphical transformation

We can see that graph has a sharp turn at +1 and -1 so the function is not differentiable at these points.