Animated Solution for Mathematics - Limits, Continuity and Differentiability: Let $f(x)=15-|x-10|;x\in \mathbb{R}$. Then the set of all values of $x$, at which the function, $g(x)=f(f(x))$ is not differentiable, is :

We are tasked with finding the points of non-differentiability for the composite function $g(x)=f(f(x))$, where the base function is defined as:

The function $f(x)$ represents an inverted V-shape with a vertex at $x=10$. At this point, the slope changes abruptly, making $f(x)$ non-differentiable at $x=10$.

To analyze $g(x)=f(f(x))$, we substitute the expression for $f(x)$ into itself:

Substituting $f(x)=15-|x-10|$ into the equation, we obtain:

Simplifying the constants inside the absolute value, we arrive at the final form of the composite function:

A function involving absolute values is non-differentiable at points where the argument of any absolute value term becomes zero. We must examine both the inner and outer layers of the nested expression.

Setting the inner argument to zero:

Since $f(x)$ is non-differentiable at $x=10$, the composite function $g(x)$ is also non-differentiable at this point.

Setting the outer argument to zero:

This simplifies to:

This equation yields two distinct solutions:

By evaluating the conditions where the arguments of the absolute value functions vanish, we have identified the complete set of points where the function $g(x)$ fails to be differentiable.

The points of non-differentiability are:

These three points represent the "sharp corners" of the composite function, where the graph undergoes sudden changes in direction.