Question

For what values of x is `f(x)=-x^2+xe^x` concave or convex?

Answer 1

The function is concave up or convex for `x>0` and concave for `x<0`

Explanation:

`f(x)=-x^2+xe^x`
`f'(x)=-2x+e^x+xe^x`
`f''(x)=-2+2e^x+xe^x`
`f''(x)=0`
`0=-2+e^x(2+x)|+2`
`2=e^x(2+x)`
if `x=0`
`e^0=1`
`(2+0)=2`
`2*1=2`
`2=e^0(2+0)`
`f''(1)=2*e-2+e>0`
The function is concave up or convex for `x>0` and concave for `x<0`