Mathematics

Problem

The temperature

T(t)

(in °F) of a greenhouse is recorded continuously over the 12-hour period from midnight (

t=0

) to noon (

t=12

). Measurements show

T(0)=60

T(6)=55

, and

T(12)=80

. Which of the following sets of claims is guaranteed to be true? a) There exists a time

t

(0,12)

with

T(t)=90

. b) There exists a single time

t

(0,12)

with

T(t)=70

and

T'(t)=0

. c) There exists a time

t_1

(0,6)

with

T(t_1)=58

and a (possibly different) time

t_2

(6,12)

with

T(t_2)=70

. d) The minimum temperature on

(0,12)

55\,^\circ\!\text F

and occurs exactly at

t=6