Question

Let F be the function given by f(x)= 2xe^x. Determine the interval(s) on which the graph is concave down.

Answer 1

Well, took find where a function concave down you know we take the second derivative and set it equal to zero to find the points of concavity. When this is done you get:
4e^x + 2xe^x = 0
Divide by the constant '2'.
e^x(2+x) = 0
And while you're at it, also divide by e^x since this will never be zero.
We get:
x + 2 = 0
x = -2
So the intervals obtained are:
(neg. infinity, -2) & (-2, pos. infinity)
Substitute a value from each interval into the 2nd derivative of F, F'' = (x+2):
1. The first interval yields a negative value.
2. The second yields a positive value.
So the interval that concave down is the one that yields the negative value:
(negative infinity, -2)