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

[carlodelmundo]

Homework Statement

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

Homework Equations

Distance Formula

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!

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.

(and minimising the inner quantity x² + e^2x is the same as minimising √(x² + e^2x))

 

[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).