Mathematics

Problem

For a real parameter

k

, define the function

F_k

on the interval

[1,e]

F_k(x)=\ln x+kx

. a. Derive an expression, in terms of

k

, for the average rate of change of

F_k

[1,e]

. Simplify completely. b. Let

c_k\in(1,e)

satisfy

F_k'(c_k)=\dfrac{F_k(e)-F_k(1)}{e-1}

. Solve for

c_k

in terms of

k

and verify that

c_k

actually lies in

(1,e)

for all

k

in its domain of validity. c. Determine the value(s) of

k

for which

c_k=2

exactly. d. Find all real values of

k

such that the corresponding

c_k

obtained in part (b) belongs to the interval

(1.5,2.5)

. Justify your result analytically, showing all algebraic steps and any inequalities used.