Suppose that h is a function with h'(x) = xe^x. Find all intervals where the function h is concave up.

Question

Now, I understand that concavity is determined by the second derivative, so what my question here is, is it just taking the first derivative and then taking it again? Is it that simple?

Also, does it just involve the Product rule when taking the first derivative?
Would it be:
y' = 1e^x + xe^x *1

How would you approach a second derivative?
y'' = e^x +1e^x +xe^x*1

y'' = 2e^x+xe^x

How would you determine concavity if you can't find where y'' = 0?

Answer 1

you already have h', the first
just take one more

x e^x + e^x = e^x (x+1)

e^x always +
so where x + 1 is +

x> -1