Question

What are the points of inflection, if any, of `f(x)= x^4-6x^3`?

Answer 1

There exists a point of inflection at `(0,0)`

Explanation:

Points of inflections exist when `(d^2y)/dx^2=0` or when `f''(x)=0`
We start by differentiating the function
`f(x)=x^4-6x^3` we use `(d(x^n))/dx=nx^(n-1)`
So `f'(x)=4x^3-18x^2`
`f'(x)= 0` when `4x^3-18x^2=0`
`2x^2(2x-9=0)` => `x=0` and `x=9/2`
Then we calculate `f''(x)=12x^2-36x`
we continue with the values of x obtained above
`f''(0)=0` which is a point of inflection
`f''(9/2)=12*(9/2)^2-36*(9/2)=12*9/4-18*9=27-162=-135` which is `<0` and is a minimum