For what values of x is #f(x)=x^3-x^2e^x# concave or convex?

To determine the concavity of ( f(x) = x^3 - x^2e^x ), you need to find the second derivative, ( f''(x) ), and then examine its sign.

[ f'(x) = 3x^2 - 2xe^x - x^2e^x ]

[ f''(x) = 6x - 2e^x - 2xe^x - 2xe^x - x^2e^x ]

To simplify:

[ f''(x) = 6x - 4xe^x - x^2e^x ]

To determine concavity, we need to find where ( f''(x) ) is positive (convex) or negative (concave). Since the function is continuous, we'll find where ( f''(x) ) changes sign by solving ( f''(x) = 0 ), and then check the intervals between the zeros.

[ 6x - 4xe^x - x^2e^x = 0 ]

[ x(6 - 4e^x - xe^x) = 0 ]

[ x = 0 \text{ or } 6 - 4e^x - xe^x = 0 ]

We may need numerical methods to solve for ( x ) in the second equation. Once we have the zeros, we can check the intervals between them. Use the second derivative test: if ( f''(x) > 0 ), the function is convex; if ( f''(x) < 0 ), it is concave.