Concave question? (calc)?

Question

On what interval is the function f(x) = x^2 e^x concave downward?

Right, but it wants two intervals. i.e. (__,__)

The answer to both is not infinity, or -infinity, nor is it zero.

im confused.

By interval given, do you mean - infinity
That is the same as saying (-infinity, 0) right?

that answer doesnt work :(

Answer 1

I don't know what that answerer above is talking about.

Everyone can easily take the derivative twice.

You do get e^x (x² + 4x + 2)

e^x is always positive, so the question is:

when is x² + 4x + 2 negative?

Since it's a parabola opening upwards, you know that's between the two roots. The quadratic equation gives us those roots:

-2±√(2)

Answer:

( -2-√(2) , -2+√(2) )

Answer 2

- infinity f'(x) = x^2e^x+2xe^x
f"(x) = x^2e^x + 2xe^x +2xe^x+2e^x=x^2e^x+4xe^x+2e^x
f"(x) = e^x(x^2 +4x+2)
f"(x) is negative when x<0
Hence f(x) is concave downward in the interval given.

Check on a graphing and you will see that my answer is correct.