Mathematics

Problem

A continuous function

h

[0,6]

is differentiable on

(0,6)

and satisfies

h(0)=h(3)=h(6)=10

. In addition,

h(1)=8

and

h(5)=12

. No further explicit formula is known. a. Part 1: (a) Explain why there must be at least one value

c_{1}

(0,3)

with

h'(c_{1})=0

and at least one value

c_{2}

(3,6)

with

h'(c_{2})=0

. b. Part 2: (b) Prove that

h

has at least three stationary points in the open interval

(0,6)

. Your argument should combine the conclusions from part (a) with appropriate reasoning about the overall behavior of

h

[0,6]

. c. Part 3: (c) Suppose further that

h''(x) < 0

for all

x

(2,4)

. Explain what this additional information implies about any stationary point that lies in that subinterval.