A function is defined on the closed interval with the following properties

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

A function

r

is defined on the closed interval

[-3,5]

with the following properties. •

r

is continuous on

[-3,1)

and on

(1,5]

. •

\displaystyle\lim_{x\to1^-}r(x)=4

and

\displaystyle\lim_{x\to1^+}r(x)=4

. •

r(1)=3

. •

r(-3)=2

and

r(5)=6

. • For

-3\le x<1

the graph of

r

rises steadily from

( -3,2)

to approach the point

(1,4)

, and for

1<x\le5

the graph rises from

(1,4)

to the endpoint

(5,6)

without turning. Based on this information, answer the following questions about the behavior of

r

on the interval

[-3,5]

. (a) Explain whether the Extreme Value Theorem can be applied to

r

on the interval

[-3,5]

. Clearly reference any required hypotheses in your explanation. (b) Determine whether

r

attains an absolute maximum on

[-3,5]

. If it does, state the maximum value and the

x

-value(s) at which it occurs; if it does not, provide a justification. (c) Determine whether

r

attains an absolute minimum on

[-3,5]

. If it does, state the minimum value and the

x

-value(s) at which it occurs; if it does not, provide a justification. (d) Describe one specific change you could make to the definition of

r

x=1

that would allow the Extreme Value Theorem to guarantee both an absolute maximum and an absolute minimum on

[-3,5]

. Explain why your change works.