Question

Is `f(x)=-4x^5-2x^4-2x^3+42x-30` concave or convex at `x=0`?

Answer 1

Neither of the two.

Explanation:

You need to check the second derivative: if it's positive in `x=0`, the function is convex. Otherwise, it's concave. So, we have

`f(x)=-4x^5-2x^4-2x^3+42x-30`

`f'(x)=-20x^4-8x^3-6x^2+42`

`f''(x)=-80x^3-24x^2-12x`

Unfortunately, `f''(0)=0`. Since it is positive before zero and negative after zero, it means that in `x=0` the function passes from being convex to being concave.