For a positive parameter

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

Problem

k

, define the function

r_k(x)=e^{x}-kx

for

x\ge0

. (a) Evaluate

r_k(0)

and simplify your expression. (b) Show that

r_k(k)>0

for all

k>0

. (c) Determine all positive values of

k

for which

r_k(x)=0

has at least one solution on the interval

[0,k]

. Justify your answer. (d) For

k=3

, explain why the solution found in part (c) must lie strictly between

0

and

3

. Include an argument ruling out solutions at the endpoints.