For , define on the interval…

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

For

k>0

, define

f_k(x)=k\ln(x)-x

on the interval

[1,e^{k}]

. (a) Derive an expression, in terms of

k

, for the average slope of

f_k

over

[1,e^{k}]

. (b) Solve

f_k'(x)=m

, where

m

is the expression from part (a), and express every solution

c

in terms of

k

. (c) Show that exactly one such value

c

lies in

(1,e^{k})

for every

k>0

. Justify your conclusion with appropriate reasoning about the behavior of

f_k'(x)