A differentiable function f has first derivative f′. The following statements about f′ and f are often used when analyzing graphs.
(a) The function f′ is positive on an interval (a,b).
(b) The function f′ changes sign from positive to negative at x=c and f′(c)=0.
Respond to each prompt below. Answers must be written in complete sentences that reference the information given and appropriate calculus principles.
a. Explain what conclusion can be drawn about the behavior of f on the interval (a,b) based on statement (a). Include a clear verbal connection between the sign of f′ and the behavior of f. No numerical calculation is required.
b. Explain what conclusion can be drawn about the point x=c on the graph of f based on statement (b). Be explicit about the type of critical point that occurs and why the sign change in f′ supports that conclusion.