Limits, Continuity & Differentiability Question - None

Question

Let $ f(x) = 15 - |x - 10|;\; x \in R $. Then the set of all values of x, at which the function, $ g(x) = f(f(x)) $ is not differentiable, is:

Accepted Answer

Step 1: Identify where f(x) = 15 - |x - 10| is non-differentiable.

f(x) has a corner point at x = 10 where the absolute value changes sign, so f is non-differentiable at x = 10.

Step 2: For g(x) = f(f(x)) to be non-differentiable, we need:

(i) Points where f(x) is non-differentiable: x = 10

(ii) Points where f(x) = 10 (since f is non-differentiable at 10)

Step 3: Solve f(x) = 10:

15 - |x - 10| = 10

|x - 10| = 5

x - 10 = ±5

x = 15 or x = 5

Step 4: Verify these are distinct points where g fails to be differentiable:

At x = 5: f(5) = 15 - 5 = 10, and f is non-differentiable at 10

At x = 10: f(10) = 15 - 0 = 15, but f itself is non-differentiable at 10

At x = 15: f(15) = 15 - 5 = 10, and f is non-differentiable at 10

∴ Answer: {5, 10, 15}

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