A function is defined on the closed interval by p(x)= cases 5- x^2 4 , & -3 x 0, 1+ x…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A function p is defined on the closed interval [−3,4] by
p(x)=⎩⎨⎧5−4x2,1+x,−3≤x≤0,0<x≤4.
The graph of p is not provided.
(1) Part 1: (a) Verify that p is continuous on [−3,4]. Provide complete reasoning, including consideration of x=0.
(2) Part 2: (b) Explain why p must attain both an absolute maximum value and an absolute minimum value on the interval [−3,4]. Cite an appropriate theorem and discuss how its conditions are satisfied by p.
(3) Part 3: (c) Describe a systematic procedure that could be used to locate the absolute extrema of p on [−3,4] without actually carrying out any numerical calculations. Your description should cite every point or type of point that must be examined and explain why each is necessary.