Question

How do you find the maximum, minimum and inflection points and concavity for the function `y = xe^x`?

Answer 1

• `x=-1` is a local minimum and the absolute minimum of the function `y=xe^x` because `y'` changes from negative to positive at `x=-1`.
• `x=-2` is an inflection point of the function `y=xe^x` because `y''` changes from negative to positive at `x=-2`.
• `y=xe^x` is concave down (convex) on `x in (-oo,-2)`, and concave up on `x in (-2,oo)`

Explanation:

`y=(x)(e^x)`

To find maxima and minima, find where `y'` is equal to zero:
`y'=(x)(e^x)+(1)(e^x)`
`y'=(e^x)(x+1)`
`0=(e^x)(x+1)`
`0=x+1`
`x=-1`
To check whether `x=-1` is a max or a min, use the first derivative test (plug in values less than and greater than -1):
`y'(-100)="negative"`
`y'(100)="positive"`
`x=-1` is a local minima and the absolute minimum of the function `y=xe^x` because `y'` changes from negative to positive at `x=-1`.

Do the second derivative test to find inflection points and concavity:
`y'=(e^x)(x+1)`
`y''=(e^x)(1)+(e^x)(x+1)`
`y''=(e^x)(x+2)`
`0=(e^x)(x+2)`
`0=x+2`
`x=-2`
`y''(-100)="negative"`
`y''(100)="positive"`
`x=-2` is an inflection point of the function `y=xe^x` because `y''` changes from negative to positive at `x=-2`.
`y=xe^x` is concave down (convex) on `x in (-oo,-2)`, and concave up on `x in (-2,oo)`