For f(x) the local concavity/convexity is given by the sign of
(d^2f)/(dx^2)(x).
For f(x) = -x^4 + 4 x^3 - 2 x^2 - x + 5
we have (d^2f)/(dx^2)(x)=-4 + 24 x - 12 x^2.
If (d^2f)/(dx^2)(x) <0 the local curvature is qualified as convex
otherwise if (d^2f)/(dx^2)(x) > 0 concave.
Extracting the roots of
-4 + 24 x - 12 x^2= 0
we get
{(x = 1/3 (3 - sqrt[6])),(x = 1/3 (3 + sqrt[6])):}
x < 1/3 (3 - sqrt[6]) -> "convexity"
1/3 (3 - sqrt[6]) < x < 1/3 (3 + sqrt[6])->"concavity"
1/3 (3 + sqrt[6])< x->"convexity"
blue f(x), pink (d^2f)/(dx^2)(x)
![enter image source here]()
