Mathematics

Mathematics

Problem

The function f(x)={π4+tan⁡−1x,∣x∣≤112(∣x∣−1),∣x∣>1f(x) = \begin{cases} \frac{\pi}{4} + \tan^{-1} x, & |x| \leq 1 \\ \frac{1}{2}(|x| - 1), & |x| > 1 \end{cases} is (a) both continuous and differentiable on R−{−1}R - \{-1\} (b) both continuous and differentiable on R−{1}R - \{1\} (c) continuous on R−{−1}R - \{-1\} and differentiable on R−{−1,1}R - \{-1, 1\} (d) continuous on R−{1}R - \{1\} and differentiable on R−{−1,1}R - \{-1, 1\}
Intuitive explanation

This problem asks you to recognize that a piecewise function is really two different "machines" glued together at a boundary. The key insight is to check whether the pieces connect smoothly—do they meet at ∣x∣=1|x| = 1, and do their slopes match there? Understanding continuity and differentiability at the junction points tells you whether ff is merely continuous, merely differentiable, or neither, which is what "is" is asking you to determine.

Intuitive explanation

This problem asks you to recognize that a piecewise function is really two different "machines" glued together at a boundary. The key insight is to check whether the pieces connect smoothly—do they meet at ∣x∣=1|x| = 1, and do their slopes match there? Understanding continuity and differentiability at the junction points tells you whether ff is merely continuous, merely differentiable, or neither, which is what "is" is asking you to determine.

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