Convexity, concavity and inflexion points of a function

The second derivative. If a derivative f ' ( x ) of a function f ( x ) is differentiable in the point ( x₀ ), then its derivative is called the second derivative of the function f ( x ) in the point ( x₀ ) and denoted as   f '' ( x₀ ).
A function  f ( x ) is called convex in an interval ( a, b ), if a graph of the function   f ( x ) is placed in this interval  lower than a tangent line, going through any point ( x₀ ,   f ( x₀ ) ),  x₀ ( a , b ).
A function  f ( x ) is called concave in an interval ( a, b ), if a graph of the function  f ( x ) is placed in this interval  higher than a tangent line, going through any point ( x₀ ,   f ( x₀ ) ),  x₀ ( a , b ).

Sufficient condition of concavity ( convexity ) of a function.
Let a function  f ( x )  be twice differentiable ( i.e. it has the second derivative ) in an interval  ( a , b ), then: if   f '' ( x ) > 0  for any  x( a , b ), then the function  f ( x ) is concave in the interval ( a , b );  if  f '' ( x ) < 0  for any  x ( a , b ), then the function  f ( x ) is convex in the interval ( a , b ).

If a function changes a convexity to a concavity or vice versa at passage through some point, then this point is called an inflexion point an inflexion point. Hence it follows, that if the second derivative f '' exists in an inflexion point x₀ , then   f '' ( x₀ ) = 0.

E x a m p l e.

Consider a graph of the function y = x 3 : This function is concave at  x > 0 and convex at  x < 0. In fact,  y'' = 6x, but 6x > 0 at x < 0 and 6x < 0 at x < 0, hence,  y'' > 0 at x 3 0 and  y'' < 0 at  x < 0, hence it follows, that the function y = x 3 is concave at  x > 0 and convex at  x < 0. Then the point x = 0 is the inflexion point of the function y = x 3.