Functions and are continuous on

MATHEMATICS · Single correct · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Functions

f

and

g

are continuous on

[0,3]

. The values

f(0)=1

g(0)=4

f(3)=5

, and

g(3)=2

are known. A student claims there must be a number

c

(0,3)

for which

f(c)=g(c)

. Which of the following statements best justifies the student’s claim? (a) Because

f

and

g

are continuous, they must intersect at the midpoint

x=1.5

. (b)

f(x)-g(x)

is continuous on

[0,3]

, and

(f-g)(0)<0

while

(f-g)(3)>0

; by the IVT there exists

c\in(0,3)

with

(f-g)(c)=0

, i.e.

f(c)=g(c)

. (c) Since

f(0)<g(0)

and

f(3)>g(3)

, we have

(f-g)(0)>0

and

(f-g)(3)<0

, so by the IVT some

c

satisfies

f(c)=g(c)

. (d) By the Mean Value Theorem applied to

h(x)=f(x)-g(x)

, there is a

c

with

h'(c)=0

, which implies

f(c)=g(c)