How to find the inflection point of e^sqrt(x)?

I have f(x) = e^sqrt(x) and I want to find the coordinates of the inflection point highlighted in the image:
enter image source here

1 Answer
Nov 5, 2017

(1,e)

Explanation:

For #f(x) = e^sqrtx), we get

f''(x) = (e^sqrtx(sqrtx-1))/(4xsqrtx)

Note that e^sqrtx is positive for all x >= 0

and 4xsqrtx is also positive for all x >= 0.

So the sign of f''(x) is the same as the sign of sqrtx -1, which is negative for 0 <=x < 1 and positive for x > 1.

So, f''(x) changes at sign at x=1

So the inflection point is (1,e^sqrt1) = (1,e)