Let defined for

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

Problem

Let

r(x)=\ln x

defined for

1\le x\le e^2

. A straight line

s(x)=mx+c

is drawn so that its graph intersects the

x

–axis at

(4,0)

and its

y

–intercept is positive. (a) Express

s(x)

in terms of the single parameter

c>0

, and indicate the permitted range of

c

so that

s

satisfies the stated geometric conditions. (b) For a fixed value of

c

in that range, determine whether

r

and

s

must intersect somewhere on

[1,e^2]

. Support your answer by evaluating a single appropriate function at the endpoints of the interval. (c) Suppose

c

varies. Find all

c

for which

r

and

s

are guaranteed to intersect on

[1,e^2]

. Show all reasoning, including any necessary inequalities.