Let be defined on the closed interval

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives

Problem

Let

r_c(x)=\ln x - c x

be defined on the closed interval

[1,e]

, where

c

is a positive real constant and

e

is the base of the natural logarithm. Take

c=0.2

. (a) Compute

r_{0.2}'(x)

and determine all critical points in

[1,e]

for

r_{0.2}

. (b) Evaluate

r_{0.2}

at the critical point(s) found in part (a) and at the endpoints

x=1

and

x=e

. (c) State the absolute maximum and absolute minimum values of

r_{0.2}

[1,e]

, including the

x

–values where each occurs. Justify your answer.