Limits, Continuity & Differentiability
Continuity And Differentiability
nta_abhyas_2025
Grade 12
Question:
Let $f(x) = 10 - |x - 5|, x \in \mathbb{R}$, then the set of all values of $x$ at which $f(f(x))$ is not differentiable is
{0, 5, 10}
{5, 10}
{0, 5, 10, 15}
{5, 10, 15}
Step-by-Step Solution
Key Concept: Analyze compositions of absolute value functions by evaluating multiple nested expressions and identifying corner points.
$f(f(x)) = |10 - |x - 5|| - 5| = |10 - |x - 5|| - 5|$. Setting this equal to zero and solving: $|10 - |x - 5|| = 5$, which gives $x - 5 = 0$ and $|x - 5| = 5$. Points where $f(f(x))$ is non-differentiable occur at $x = 5 \& x = 0, 10$.
Correct Answer: (0, 5, 10)