Consider the piecewise–defined function g(x)= cases 2x+5, & -4 x -1, 4-(x+1)^2 +3, &…
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
Consider the piecewise–defined function
g(x)=⎩⎨⎧2x+5,4−(x+1)2+3,−x+7,−4≤x≤−1,−1<x<3,3≤x≤6.
The graph of g consists of a line segment from (−4,−3) to (−1,3), the upper half of a circle of radius 2 centered at (−1,3) from x=−1 to x=3, and another line segment from (3,4) to (6,1).
Answer the following questions about g on the interval [−4,6].
(1) Part 1: (a) Determine all x-values in [−4,6] where g′(x)=0.
(2) Part 2: (b) Determine all x-values in [−4,6] where g′(x) does not exist. Explain geometrically why the derivative fails to exist at each of those points.
(3) Part 3: (c) Using the First Derivative Test, classify each critical point found in parts (a) and (b) as a local maximum, a local minimum, or neither.
(4) Part 4: (d) Determine the absolute maximum and absolute minimum values of g on [−4,6] and the x-values at which they occur. Provide full justification, citing all relevant candidates.