A function is defined on the closed interval by p(x)= cases x, & - x<0, 1, & x=0,…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A function p is defined on the closed interval [−π,2] by
p(x)=⎩⎨⎧sinx,1,1−x,−π≤x<0,x=0,0<x≤2.
You will investigate whether absolute extrema exist for p on its domain.
(a) Determine whether p is continuous at x=0. Provide a clear limit–based justification for your conclusion.
(b) Decide whether it is mathematically valid to restrict attention only to interior points with p′(x)=0 or p′(x) undefined when looking for absolute extrema of p on [−π,2]. Explain how your answer to part (a) influences this decision.
(c) For the portions of the domain where p is differentiable, compute p′(x) and find the interior point(s) where p′(x)=0.
(d) Evaluate p at every point that could plausibly be an absolute maximum or minimum, including endpoints, interior critical points, and any other candidates justified by your analysis.
(e) State whether p attains an absolute maximum value and an absolute minimum value on [−π,2]. If so, identify the values and where they occur; if not, give a rigorous explanation for the failure.