Question

Is `f(x)=-x^5-21x^4-2x^3+4x-30` concave or convex at `x=0`?

Answer 1

inflexion point

Explanation:

Calculating the first derivative of `f(x)=-x^5-21x^4-2x^3+4x-30`
at `x=0` we get `f_x(0)=0` The second derivative gives
`f_{x x}(0) = 0` and the third `f_{x x x}(0)=-12` so the origin is an inflexion point. In this point the function behavior is like `-abs(c_0) x^3`. Nor concave or convex.