Optimization - Finding Minimum Between (0,0) and e^x

[carlodelmundo]

1. The problem statement, all variables and given/known data

Find the minimum distance from the origin to the curve y = e^x.


2. Relevant equations

Distance Formula

3. The attempt at a solution

http://carlodm.com/calc/prob6.jpg

5-6 bright Calculus kids in my high school grappled with this problem and we couldn't find an answer.

Can anyone verify my solution? To simplify calculations, I minimized the inner quantity (underneath the square root of the Distance Formula). I feel, though, that this may have changed the answer. Answers are so close that I may have made a mistake.

[tiny-tim]

Hi carlodelmundo! :smile:
Find the minimum distance from the origin to the curve y = e^x.
…
Can anyone verify my solution? To simplify calculations, I minimized the inner quantity (underneath the square root of the Distance Formula). I feel, though, that this may have changed the answer.

Looks good to me. :biggrin:

(and minimising the inner quantity x2 + e2x is the same as minimising √(x2 + e2x)) :smile:

[carlodelmundo]

okay! just checking. Thank you tiny-tim.

I thought that since the derivatives of √(x^2 + e^2x)) and x^2 + e^2x are different, there could have been a discrepancy in my answer.

[tiny-tim]

I thought that since the derivatives of √(x^2 + e^2x)) and x^2 + e^2x are different, there could have been a discrepancy in my answer.

nah … if f'(x)/2√(f(x)) = 0, then that's the same as f'(x) = 0 (unless f(x) can be infinite). :wink: