The function is continuous and differentiable on

MATHEMATICS · Single correct · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

The function

g

is continuous and differentiable on

[0,4]

. The slope of the secant line joining

(0,g(0))

and

(4,g(4))

\dfrac12

. At

x=2

, the tangent to the graph of

g

is horizontal. Which of the following must be true? (a)

g'(x)=\dfrac12

for all

x

(0,4)

. (b) There exists at least one

c

(0,4)

such that

g'(c)=\dfrac12

. (c)

g'(x)>0

for all

x

(0,4)

. (d) Because the interval midpoint is

x=2

, the Mean Value Theorem guarantees

g'(2)=\dfrac12