Question

What are the points of inflection, if any, of `f(x) = 2x^3 + 3x^2 - 432x`?

Answer 1

At inflexion points, the curve changes its concavity. We can test for these changes using the 2nd derivative, for when `f''(x)=0`. Keep in mind that this does not work for all functions (an example would be for `x^4`). Use a sign table/table of values to check if you are unsure (to see if the point is a turning point or an inflexion point).
`f'(x)=6x^2 + 6x -432`
`f''(x)=12x+6`
Solving for `f''(x)=0`
`12x+6=0`
`2x=-1`
`x=-1/2`
At `x=-1/2, f(x)=433/2`
So the point of inflexion is at `(-1/2,433/2)`