Question

What are the points of inflection, if any, of `f(x) =x/(x+8)^2`?

Answer 1

Below

Explanation:

`f(x)=x/(x+8)^2`
`f'(x)=((x+8)^2(1)-(x)2(x+8))/(x+8)^4`
`f'(x)=((x+8)-2x)/(x+8)^3`
`f'(x)=(8-x)/(x+8)^3`
`f''(x)=((x+8)^3(-1)-(8-x)(3)(x+8)^2)/(x+8)^6`
`f''(x)=((-1)(x+8)-3(8-x))/(x+8)^4`
`f''(x)=(-x-8-24+3x)/(x+8)^4`
`f''(x)=(2x-32)/(x+8)^4`

For points of inflexion, `f''(x)=0`

ie

`(2x-32)/(x+8)^4=0`
`2x-32=0`
`x=16`

Test `x=16`

When:
`x=15`, `f''(x)=-2/279841`
`x=16`, `f''(x)=0`
`x=17`, `f''(x)=2/390625`

Since there is a change in concavity, then there is a point of inflexion at `(16,1/36)`