Question
Given the graph of f ′, find the following properties of the function f :
(A) The intervals on which f is increasing or decreasing
(B) The location of the relative maxima and minima
(C) The points of inflection and concavity of f
(D) Draw a sketch of f , given that f (-1) = f (1) = 5, f (0) = 0, and f (5) = -5.
Answer/Explanation
(A) f ′ < 0 on (-∞, -1) and (1, 5), so f is decreasing. f ′ > 0 on (-1, 1) and (5, +∞), so f is increasing.
(B) The function has a relative maximum at x = 1 since f ′ changes from positive to negative. There are relative minima at x = (-1, 5) since f ′ changes from negative to positive.
(C) f ′ is increasing on (-∞, 0), so f ′′ > 0 and f is concave upward. f ′ is decreasing on \(\left ( 0,2\frac{1}{2} \right )\) so f ′′ < 0 and f is concave downward. f ′ is increasing on\(\left ( 2\frac{1}{2} ,+\infty \right ) \), so f ′′ > 0 and f is concave upward. A change of concavity occurs at x = 0 and \(x=2\frac{1}{2}\) .Summarizing the results in a table: