Barrons AP Calculus

(I)

(II)

(A)

(B)

(C)

(D)

(E)

,

it follows that F(x) > 0 if 4 < x < 6.
(ii) is false. Since F ′(x) = f(x) and f(x) < 0 if 4 < x < 5, then F ′(x) < 0.
(iii) is false. Since F ′(x) = f(x), F′′(x) = f ′(x). At x = 6, f ′(x) > 0 (because f is
increasing). Therefore, F′′(6) > 0.

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:

f(x)=   g′(x)

h(x)    =   f   ′(x)

(III)

Which of these statements is (are) true?

I   only

II  only

III only

all three

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.