# Second derivative of sqrt(x) * e^(-x)

## Homework Statement

find f''(x) if f(x) = sqrt(x) * e^(-x) and then find the roots of f''(x)

// im trying to do the 2nd derivative test (need f''x) and then find inflection points//

## Homework Equations

my methodology| d/dx sqrt(x) = 1/(2*sqrt(x)) and d/dx e^(-x) = -e^(-x)

## The Attempt at a Solution

i found f'(x) to be: e^(-x) /(2*sqrt(x)) + (-e^(-x) *sqrt(x))

and then f''(x) should be d/dx [ e^(-x) /(2*sqrt(x))] + d/dx [-e^(-x) *sqrt(x)]

and ive gone through it a few times but what i get is:

-e^(-x)*(2*sqrt(x)) - [2/(2*sqrt(x))] + [e^(-x)*sqrt(x) + -e^(-x)/(2*sqrt(x))]

when i try to set this to zero i just get the feeling that my f'' is just completely wrong. When i first saw this problem i thought "no problem" but now i dont know. Is this problem covertly difficult, or am i doint something wrong? THANKS!

## Answers and Replies

Looks like your f'(x) is right.

However, yeah, I'd say your f''(x) has some issues.

Remember that 1/sqrt(x) = x^(-1/2). If you remember that, it should be easier to take your derivative, just remember to do the product rule again.

This means that d/dx (1/sqrt(x)) = d/dx (x^(-1/2)) = -(1/2)x^(-3/2).

Hope that helps a bit!