Key Concepts and Formulas

• Differentiability of Absolute Value Functions: The function $|u(x)|$ is not differentiable at points where $u(x) = 0$, provided that $u'(x)$ exists at those points. The sharp turns in the graph of the absolute value function lead to non-differentiability.
• Composition of Functions: If $g(x) = f(h(x))$, then the differentiability of $g(x)$ depends on the differentiability of $f$ and $h$. Specifically, if $h(x)$ is not differentiable at some point $a$, then $g(x)$ might not be differentiable at $a$. If $f(u)$ is not differentiable at $u = h(a)$, then $g(x)$ might not be differentiable at $a$.
• Properties of Modulus Function: $|a| = a$ if $a \ge 0$, and $|a| = -a$ if $a < 0$. Also, $|a| = |b|$ implies $a = b$ or $a = -b$.

Step-by-Step Solution

Step 1: Analyze the inner function $f(x)$ The given function is $f(x) = 15 - |x - 10|$. The absolute value function $|x - 10|$ has a point of non-differentiability where its argument is zero, i.e., $x - 10 = 0$, which means $x = 10$. The function $f(x)$ is a transformation of $|x - 10|$. Specifically, it's a reflection about the x-axis and a vertical shift. The graph of $f(x)$ will have a peak at $x=10$, where $f(10) = 15 - |10 - 10| = 15$. For $x > 10$, $f(x) = 15 - (x - 10) = 25 - x$. For $x < 10$, $f(x) = 15 - (-(x - 10)) = 15 + x - 10 = 5 + x$. The function $f(x)$ is differentiable everywhere except at $x = 10$.

Step 2: Define the composite function $g(x)$ We are given $g(x) = f(f(x))$. Substituting the expression for $f(x)$ into itself: $g(x) = f(f(x)) = 15 - |f(x) - 10|$

Step 3: Substitute the expression for $f(x)$ into $g(x)$ $g(x) = 15 - |(15 - |x - 10|) - 10|$ $g(x) = 15 - |15 - |x - 10| - 10|$ $g(x) = 15 - |5 - |x - 10||$

Step 4: Identify potential points of non-differentiability for $g(x)$ The function $g(x)$ is of the form $15 - |h(x)|$, where $h(x) = 5 - |x - 10|$. The function $|h(x)|$ is not differentiable where $h(x) = 0$. So, we need to find the values of $x$ for which $5 - |x - 10| = 0$. This implies $|x - 10| = 5$.

Step 5: Solve for $x$ when $h(x) = 0$ The equation $|x - 10| = 5$ means: $x - 10 = 5$ or $x - 10 = -5$. Solving these, we get: $x = 10 + 5 \Rightarrow x = 15$ $x = 10 - 5 \Rightarrow x = 5$ So, $g(x)$ might be non-differentiable at $x = 5$ and $x = 15$.

Step 6: Consider the differentiability of the inner absolute value function within $h(x)$ The expression $h(x) = 5 - |x - 10|$ itself contains an absolute value function, $|x - 10|$. The function $|x - 10|$ is not differentiable at $x - 10 = 0$, which is $x = 10$. Therefore, $h(x) = 5 - |x - 10|$ is also not differentiable at $x = 10$. Since $g(x) = 15 - |h(x)|$, and $h(x)$ is not differentiable at $x = 10$, $g(x)$ will also be non-differentiable at $x = 10$.

Step 7: Consolidate all points of non-differentiability From Step 5, we found potential points of non-differentiability at $x = 5$ and $x = 15$ where the argument of the outer absolute value function becomes zero. From Step 6, we found a point of non-differentiability at $x = 10$ due to the inner absolute value function. Thus, the set of all values of $x$ at which $g(x)$ is not differentiable is $\{5, 10, 15\}$.

Common Mistakes & Tips

• Forgetting the inner points of non-differentiability: When dealing with composite functions involving absolute values, it's crucial to consider the points where the inner functions are not differentiable, not just where the outer function's argument becomes zero.
• Incorrectly solving absolute value equations: Ensure that $|a|=b$ (where $b>0$) is solved as $a=b$ or $a=-b$.
• Assuming differentiability of $f(x)$ everywhere: The function $f(x)$ is only differentiable for $x \neq 10$. This must be taken into account when analyzing $g(x)=f(f(x))$.

Summary

To find the points of non-differentiability for $g(x) = f(f(x))$, we first expressed $g(x)$ explicitly in terms of $x$. We found $g(x) = 15 - |5 - |x - 10||$. The non-differentiability of an absolute value function of the form $|u(x)|$ occurs when $u(x) = 0$. Therefore, we set the argument of the outer absolute value to zero: $5 - |x - 10| = 0$, which led to $|x - 10| = 5$, yielding $x = 5$ and $x = 15$. Additionally, we recognized that the inner absolute value function $|x - 10|$ is not differentiable at $x = 10$. Since $f(x)$ is not differentiable at $x=10$, and $g(x)$ involves $f(f(x))$, we must also consider $x=10$ as a point of non-differentiability for $g(x)$. Combining these, the set of all values of $x$ where $g(x)$ is not differentiable is $\{5, 10, 15\}$.

The final answer is \boxed{{5,10,15}}.