Consider the piecewise function f(x)= cases 1 x+2 , & -2<x 0, ( x), & 0<x 1

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives

Problem

Consider the piecewise function f(x)={1x+2,−2<x≤0,sin⁡(πx),0<x≤1.f(x)=\begin{cases} \dfrac{1}{x+2}, & -2<x\le 0,\\ \sin(\pi x), & 0<x\le 1. \end{cases} Answer the following questions about ff on the intervals shown. (1) Part 1: Determine whether the hypotheses of the Extreme Value Theorem are satisfied on the interval [−2,1][-2,1]. Justify your conclusion carefully. (2) Part 2: Determine whether the hypotheses of the Extreme Value Theorem are satisfied on the interval (−2,1](-2,1]. Justify your conclusion carefully. (3) Part 3: For the interval in part (a), identify all points at which ff attains an absolute maximum or an absolute minimum, and state the corresponding extremum values. (4) Part 4: Explain why any absolute extremum identified in part (c) must occur either at an endpoint of the interval or at a critical point of ff.

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