Question

What are the points of inflection, if any, of `f(x) = x^5/20 - 5x^3 + 5`?

Answer 1

The points of inflection are `x= -sqrt30, 0, and sqrt(30)`.

Explanation:

`f(x) = (x^5/20) - 5x^3 + 5`

Points of inflection are where the second derivative of a function is equal to 0, and where the second derivative of a function switches signs. So first you find the second derivative and set it equal to 0.

`f'(x) = (5x^4/20) - 15x^2 = (x^4/4) -15x^2`
`f''(x) = (4x^3/4) - 30x = x^3 - 30x`
`x^3 - 30x = 0`

Factor out the x.
`x(x^2 - 30) = 0`
so `x=0` and `x = +- sqrt(30)`

To check if these are all inflection points, you check if the signs change around them. I plugged in `x=-6, -1, 1`, and `6`, but you can do this with any points outside and between the three answers. The signs changed around all 3 points, so all 3 are points of inflection.