Question

How do you find the inflection point of the function `f(x)=xe^(-2x)`?

Answer 1

Below

Explanation:

You can find the inflexion point of a function by first finding the second derivative and letting the equation equal to 0.

`f(x)=xe^(-2x)`
`f'(x)=xtimes-2e^(-2x)+e^(-2x)`
`f'(x)=e^(-2x)(1-2x)`
`f''(x)=e^(-2x)times(-2)+(1-2x)(-2e^(-2x))`
`f''(x)=-2e^(-2x)-2e^(-2x)+4xe^(-2x)`
`f''(x)=4xe^(-2x)-4e^(-2x)`

To find the inflexion point, `f''(x)=0`

`4xe^(-2x)-4e^(-2x)=0`
`4e^(-2x)(x-1)=0`
`4e^(-2x)=0` or `x-1=0`

`4e^(-2x)=0` has no solution because `e^(-2x) >0`

`x-1=0`
`x=1`

Test `x=1` (YOU MUST TEST YOUR POINT FOR CONCAVITY)

When `x=0`,
`f''(0)=-4`
When `x=1`,
`f''(1)=0`
When `x=2`,
`f''(2)=0.07326...`

Therefore, there is a change in concavity and so there is a point of inflexion at `(1,e^(-2))`