A car’s odometer reading (in kilometers) is modeled by a continuous function of time…

MATHEMATICS · Single correct · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

A car’s odometer reading

D(t)

(in kilometers) is modeled by a continuous function of time

t

(in hours). The recorder shows

D(2)=125.3

and

D(5)=227.8

. Which statement most strongly supports the claim that, at some time

c

between 2 and 5 hours, the odometer read exactly

200

kilometers? (a) Because the car’s speed never becomes negative during the trip, the Mean Value Theorem guarantees

D(c)=200

for some

c

(2,5)

. (b) Since

D

is continuous on

[2,5]

and

200

lies between

125.3

and

227.8

, the Intermediate Value Theorem ensures

D(c)=200

for some

c

(2,5)

. (c) Because

D(5)-D(2)\approx102.5

, the average speed is constant, so

D

must pass through every value between

125.3

and

227.8

. (d) Since

D

increases by more than

100

\,km over

[2,5]

, the Extreme Value Theorem implies that

D

must equal

200

at both an absolute maximum and an absolute minimum in the interval.