Let be the function defined by q(x)= cases x^2-1, & x<1

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Let

q

be the function defined by

q(x)=\begin{cases} x^2-1, & x<1,\\ mx+b, & 1\le x\le 4,\\ \sqrt{9-(x-4)^2}, & 4<x\le 7. \end{cases}

The constants

m

and

b

are real numbers. (a) Find the values of

m

and

b

that make

q

continuous at both

x=1

and

x=4

. (b) Using your values of

m

and

b

, verify that

q

is continuous on the entire interval

[1,7]

. (c) Show that the equation

q(x)=2

has at least one real solution in the interval

(1,7)

. Give reasoning that identifies an interval of length

1

within

(1,7)

that must contain such a solution.