Design a real–valued function that satisfies all of the following conditions
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Design a real–valued function p that satisfies all of the following conditions.
• p is continuous on the closed interval [−3,3].
• p is differentiable on (−3,3) except possibly at x=0.
• x=0 is a critical point of p that is not a local extremum.
• p attains an absolute maximum at x=3 and an absolute minimum at x=−3.
(a) Provide an explicit formula (an equation or a clearly defined piecewise rule) for such a function p. State any constants you introduce.
(b) Justify that your formula meets the continuity requirement on [−3,3]. Include any necessary limit calculations.
(c) Verify the differentiability conditions on (−3,3), specifying precisely what happens at x=0.
(d) Demonstrate that x=0 is a critical point of p and explain in detail why it fails to be either a local maximum or a local minimum.
(e) Confirm that p achieves its absolute extremum values at x=−3 and x=3, and nowhere else on the interval.