Question

What are the points of inflection, if any, of `f(x)=x^4-x^2+5`?

Answer 1

Inclection points are `(sqrt6/6,187/36)` and `(-sqrt6/6,187/36)`.

Explanation:

`color(white)xxf(x)=x^4-x^2+5`

`=>d/dx[f(x)]=4x^3-2x`
`=>d^2/dx^2[f(x)]=12x^2-2`

For `d^2/dx^2[f(x)]=0`,
`color(white)xx12x^2-2=0`

`=>x=+-sqrt6/6`

For `x=sqrt6/6`,
`color(white)xxy=color(red)((sqrt6/6))^4-color(red)((sqrt6/6))^2+5`
`color(white)(xxx)=1/6^2-1/6+5`
`color(white)(xxx)=187/36`

For `x=-sqrt6/6`,
`color(white)xxy=187/36`

Therefore inclection points are `(sqrt6/6,187/36)` and `(-sqrt6/6,187/36)`.