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

Question

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

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Answer 1 · 2017-11-05T18:03:28

#(1,e)#

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)#