The function is the first derivative of a differentiable function
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
The function g is the first derivative of a differentiable function f. A verbal description of the graph of g is provided.
• g(x)=0 at x=−3, x=0, and x=4.
• g(x)<0 for x<−3.
• g(x)>0 for −3<x<0.
• g(x)<0 for x<4 and x>0 is a misprint. However, according to the original text, it should read: g(x)<0 for 0<x<4.
• g(x)>0 for x>4.
• g is continuous everywhere.
Based solely on this information about g, analyze the behavior of f.
(1) Part 1: (a) Determine the open intervals on which f is increasing and the open intervals on which f is decreasing. Explain your reasoning.
(2) Part 2: (b) Identify all x-coordinates of local extrema of f and state whether each is a local maximum or a local minimum. Justify each classification.
(3) Part 3: (c) Describe how the concavity of f might behave near x=−3, x=0, and x=4, even though no information about g′ is given. Your answer should be qualitative rather than computational.