Question

If the function f is defined as $f\left( x \right)=15-\left| x-10 \right|$;$x\in R$, then find the set of all values of x at which the function $g\left( x \right)=f\left( f\left( x \right) \right)$ is not differentiable?
(a) $\left\{ 5,10,15,20 \right\}$,
(b) $\left\{ 10,15 \right\}$,
(c) $\left\{ 5,10,15 \right\}$,
(d) $\left\{ 10 \right\}$.

Answer 1

Hint: We start solving the problem by finding the function $g\left( x \right)=f\left( f\left( x \right) \right)$. We use the fact that the function $\left| x-a \right|$ is not differentiable at $x=a$ to get our first non-differentiable point. Similarly, we get a function that resembles the function $\left| x-a \right|$ later while solving the problem to get two more non- differentiable points to get the required solution set for x.

Complete step-by-step answer:
According to the problem, we have defined function f as $f\left( x \right)=15-\left| x-10 \right|$; $x\in R$. We need to find the set of all values of x at which the function $g\left( x \right)=f\left( f\left( x \right) \right)$ is not differentiable.
Let us first find the function $g\left( x \right)$. We have $g\left( x \right)=f\left( f\left( x \right) \right)$.
$\Rightarrow g\left( x \right)=f\left( 15-\left| x-10 \right| \right)$.
$\Rightarrow g\left( x \right)=15-\left| 15-\left| x-10 \right|-10 \right|$.
$\Rightarrow g\left( x \right)=15-\left| 5-\left| x-10 \right| \right|$.
We have found the function ‘g’ as $g\left( x \right)=15-\left| 5-\left| x-10 \right| \right|$.
We know that the function $\left| x-a \right|$ is not differentiable at $x=a$. Using this fact, we can say that $f\left( x \right)=15-\left| x-10 \right|$ is not differentiable at $x=10$ as the function $\left| x-10 \right|$ is not differentiable at $x=10$. This makes the function $g\left( x \right)=15-\left| 5-\left| x-10 \right| \right|$ is not differentiable at $x=10$ ---(1).
Now, we can see that the function $g\left( x \right)=15-\left| 5-\left| x-10 \right| \right|$ has another function $\left| 5-\left| x-10 \right| \right|$ which not differentiable at $\left| x-10 \right|=5$. Now, we find the absolute values at which $\left| x-10 \right|=5$ holds true.
We know that $\left| x-a \right|=\left\{ \begin{matrix}
   x-a,\text{ if }x > a \\
   -\left( x-a \right),\text{ if }x < a \\
\end{matrix} \right.$. Using this fact, we get $x-10=5$ and $x-10=-5$.
$\Rightarrow x=5+10$ and $x=-5+10$.
$\Rightarrow x=15$ and $x=5$.
So, we get the function $\left| 5-\left| x-10 \right| \right|$ not differentiable at $x=15$ and $x=5$ which makes the function $g\left( x \right)=15-\left| 5-\left| x-10 \right| \right|$ not differentiable at $x=15$ and $x=5$ ---(2).
From equations (1) and (2), we can see that the function $g\left( x \right)=15-\left| 5-\left| x-10 \right| \right|$ is not differentiable at $x=10$, $x=15$ and $x=5$.
We have found the set of all values of x at which $g\left( x \right)=15-\left| 5-\left| x-10 \right| \right|$ is not differentiable as $x=\left\{ 5,10,15 \right\}$.
∴ The set of all values of x at which $g\left( x \right)=15-\left| 5-\left| x-10 \right| \right|$ is not differentiable as $x=\left\{ 5,10,15 \right\}$.
The correct option for the given problem is (c).

Note: We can alternatively solve by drawing the graph of the function $g\left( x \right)=15-\left| 5-\left| x-10 \right| \right|$ and finding the sharp edges in that curve.

We can see that the graph has sharp edges at $x=5$, $x=10$ and $x=15$ which makes the given function $g\left( x \right)=15-\left| 5-\left| x-10 \right| \right|$ non-differentiable at those points. Whenever we see the problems to find the non-differentiable points, it is recommended to draw a graph and check the values of x at which the sharp edges occur.