Let be the function defined on the closed interval by g_k(x)=e^ -x +k x

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Let

g_k

be the function defined on the closed interval

[0,3]

g_k(x)=e^{-x}+k x,

where

k

is a real parameter. (a) Compute

g_k'(x)

and show that

g_k

has exactly one critical point in

[0,3]

. Express the

x

–coordinate of this critical point in terms of

k

. (b) Determine the set of all

k

for which the critical point obtained in part (a) lies strictly between

0

and

3

. (c) For those values of

k

identified in part (b), compare

g_k

evaluated at the critical point with its values at the endpoints

x=0

and

x=3

. Establish precisely the range of

k

for which the absolute minimum of

g_k

[0,3]

occurs at the critical point rather than at an endpoint. Provide full justification, including any necessary inequalities.