Limits, Continuity & Differentiability Question - None

Question

Let $$f(x) = \begin{cases} \max\{|x|, x^2\}, & |x| \leq 2 \ 8 - 2|x|, & 2 < |x| \leq 4 \end{cases}$$ Let S be the set of points in the interval $(-4, 4)$ at which f is not differentiable. Then S:

Accepted Answer

Step 1: Identify the piecewise structure and boundaries in (-4, 4):

For |x| ≤ 2: f(x) = max{|x|, x²}
For 2 < |x| < 4: f(x) = 8 - 2|x|

Step 2: Analyze max{|x|, x²} on [-2, 2]:

For |x| ≤ 1: x² ≤ |x|, so f(x) = |x|
For 1 < |x| ≤ 2: |x| < x², so f(x) = x²
At x = ±1: transition points where |x| = x²

Step 3: Check differentiability at x = 1:

Left derivative (from x²): lim(h→0⁻) [2(1+h)] = 2
Right derivative (from |x|): lim(h→0⁺) [1] = 1
Left ≠ Right, so non-differentiable at x = 1

Step 4: By symmetry, non-differentiable at x = -1

Step 5: Check x = ±2 (boundary of first piece):

At x = 2: First piece gives f(x) = x² with f'(2⁻) = 4; second piece gives f(x) = 8 - 2|x| with f'(2⁺) = -2
4 ≠ -2, so non-differentiable at x = 2
By symmetry, non-differentiable at x = -2

Step 6: Check continuity and differentiability elsewhere:

f is continuous everywhere on (-4, 4)
Second piece f(x) = 8 - 2|x| is differentiable for x ∈ (2, 4) ∪ (-4, -2) except at x = 0 in the second piece (but x = 0 is in first piece)

∴ S = {-2, -1, 1, 2} (four points)

Question:

Step-by-Step Solution

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