Mathematics

Problem

For a positive parameter

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.

Intuitive explanation

The key is to understand how the exponential $e^x$ and the linear term $kx$ compete as $x$ grows. For small $x$ , the line might dominate, but exponentials always eventually win—the question is where and how. Finding the minimum of $r_k$ reveals the critical balance point: where the slope of $e^x$ (which is $e^x$ itself) equals the constant slope $k$ . This minimum tells you whether $r_k$ stays positive, touches zero, or dips below it, which determines the behavior of the entire function.

Intuitive explanation

Browse the problem library