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{cases}g_1(x), & x<a \\ g_2(x), & x \geq a\end{cases}
$
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{cases}\left(g_1(x)\right)^{\prime}, & x<a \\ \left(g_2(x)\right)^{\prime}, & x>a\end{cases}
$
Differentiability can be checked at $\mathrm{x}=\mathrm{a}$ by comparing

$
\lim _{x \rightarrow a^{-}}\left(g_1(x)\right)^{\prime} \text { and } \lim _{x \rightarrow a^{+}}\left(g_2(x)\right)^{\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{aligned}
& \mathrm{f}(\mathrm{x})=\left\{\begin{array}{cl}
-\sin x, & x<0 \\
\sin x, & x \geq 0
\end{array}\right. \\
& \therefore \quad \\
& \therefore \quad \mathrm{f}^{\prime}(\mathrm{x})=\left\{\begin{array}{cc}
-\cos x, & x<0 \\
\cos x, & x>0
\end{array}\right.
\end{aligned}
$
$
\therefore \quad \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)=\|\log |x|\|, 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.