Mathematics

Problem

Let the function

g

be defined on the closed interval

[-1,3]

g(x)=\begin{cases} x^{2}, & -1\le x\le 1,\\ 2x+1, & 1<x\le 3. \end{cases}

Answer the following questions about

g

and its behavior on

[-1,3]

. a. Part 1: (a) Determine whether

g

is continuous at

x=1

. Show analytic work to support your conclusion. b. Part 2: (b) Calculate the average value of

g

on the interval

[-1,3]

. c. Part 3: (c) State whether there must exist at least one number

c

[-1,3]

for which

g(c)

equals the average value found in part (b). Justify your answer, and find all such numbers

c

. d. Part 4: (d) Discuss how the shape of the graph of

g

helps explain the number of solutions you obtained in part (c).