Question

How do you find the inflection points of the graph of the function: `y=1/3x^3`?

Answer 1

`(x,y)=(0,0)` is an inflection point.

Explanation:

`y=1/3x^3`

`frac{d^2y}{dx^2}=2x`

At inflection point, second derivative must be zero, which occurs at `x=0`.

`frac{d^2y}{dx^2}|_{x=0}=2(0)=0`

Now, do the first derivative test.

`frac{dy}{dx}=x^2` changes from positive to zero to positive in a small neighborhood around `x=0`. `x=0` is an inflection point.