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**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.

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