Suppose a continuous function satisfies ,

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

Problem

Suppose a continuous function

f

satisfies

f(1)= -1

f(1.5)=0.875

, and

f(2)=5

. After sampling these values, a student asserts that

f

has at least one real zero in the open interval

(1,1.5)

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

f(1)\,f(1.5)<0

, the Mean Value Theorem guarantees a zero in

(1,1.5)

. (b) Because

f(1)

and

f(1.5)

have opposite signs, the Intermediate Value Theorem guarantees a root in

(1,1.5)

. (c) Because

f(1)

is negative and

f(2)

is positive, a root lies in

(1,2)

; therefore it must also lie in the subinterval

(1,1.5)

. (d) Because the average value

\dfrac{f(1)+f(1.5)}{2}

is negative, the graph must cross the

x

-axis somewhere between

1

and

1.5