Tailor Expansion

Tailor Expansion is an approximation of a function

F(t) at t = 0 is approximated as

F(0) + F'(0)t + F''(0)t^2/2! + F'''(0)t^3/3!.....

We usually calculate until second derivative as the higher derivatives are more

At classification of stationary points, the first derivative is canceled out as stationary points are equal to when F'(t) = 0 as a result: F(0) + F''(0)t^2/2!

At a certain point x = a, the Tailor expansion is

F(0) + F'(0)(x-a) + F''(0)(x-a)^2/2! + F'''(0)(x-a)^3/3!.....

The quadratic term if negative shows that at the point the function has a maximum. If positive then it's a minimum.

Convex and Concave functions

When F''(x) > 0 for all x, the function is convex on the interval. Convex function lies on or above all the tangent lines. If convex has a local minimum at x = a, it is a global minimum.

When F''(x) < 0 for all x, the function is concave on the interval. Concave function lies on or below all the tangent lines, If a concave has a local maximum at x = a, it is a global maximum.