EXAMPLE 32
Graphs of functions f (x), g(x), and h(x) are given in Figures N6–15a, N6–15b, and N6–15c.
Consider the following statements:
(I) f (x) = g ′(x) (II) h(x) = f ′(x) (III)
Which of these statements is (are) true?
(A) I only
(B) II only
(C) III only
(D) all three
(E) none of them
SOLUTION:
The correct answer is D.
I is true since, for example, f (x) = 0 for the critical values of g: f is positive where g increases,
negative where g decreases, and so on.
FIGURE N6–15a
II is true for similar reasons.
III is also true. Verify that the value of the integral g(x) increases on the interval −2.5 < x < 0
(where f > 0), decreases between the zeros of f (where f < 0), then increases again when f
becomes positive.
FIGURE N6–15b