A differentiable function satisfies the following conditions on the open interval
MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change
Problem
A differentiable function f satisfies the following conditions on the open interval (0,8).
• f′(x)>0 for 0<x<3.
• f′(x)=0 at x=3 and x=5.
• f′(x)<0 for 3<x<5.
• f′(x)>0 for 5<x<8.
Define
g(x)=∫0xf(t)dt
for 0≤x≤8.
(a) Prove that, on the interval (0,8), g can have points of inflection only at x=3 or x=5.
(b) Determine at which of the points x=3 and x=5 the graph of g actually has a point of inflection. Provide a complete justification that addresses each point separately.
(c) Explain why no additional points of inflection of g can occur on (0,8) under the given hypotheses.