A function is continuous on the interval and satisfies the following sign pattern
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A function f is continuous on the interval [−4,6] and satisfies the following sign pattern.
• f(x)>0 for −4<x<−1 and 2<x<5
• f(x)=0 at x=−1,2,5
• f(x)<0 for −1<x<2 and 5<x<6
Let the accumulation function g be defined by g(x)=∫0xf(t)dt for all x in [−4,6].
(a) Determine the intervals on [−4,6] where g is increasing and the intervals where g is decreasing. Provide a brief explanation that references the given information about f.
(b) Identify all x–values in (−4,6) at which g has a relative maximum or a relative minimum. For each identified value, classify the type of extremum and justify your conclusion.
(c) Decide whether any absolute maximum or absolute minimum of g on [−4,6] must occur at an endpoint of the interval. Support your answer with reasoning based on the given sign pattern of f.