Define the function s(x)=(x-1)^ 1/5 (x+1)^ 2/5

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

Define the function s(x)=(x−1)1/5(x+1)2/5,s(x)=(x-1)^{1/5}(x+1)^{2/5}, where real exponents denote the real fifth roots. The domain of ss is all real numbers. (a) Verify that ss is continuous at x=1x=1 and at x=−1x=-1. (b) For x≠±1x\neq\pm 1, find an explicit expression for s′(x)s'(x) by applying appropriate differentiation rules. (c) Compute lim⁡x→1s′(x)\displaystyle\lim_{x\to 1}s'(x) and lim⁡x→−1s′(x)\displaystyle\lim_{x\to -1}s'(x), if they exist. Show algebraic simplification that leads to your conclusions. (d) Classify the nature of the nondifferentiability at each of x=1x=1 and x=−1x=-1 as either a sharp corner, cusp, or vertical tangent. Provide complete justification based on your previous work.

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