Convexity, concavity and inflexion points of a function
The second derivative. Convex and concave function.
Sufficient condition of concavity ( convexity ) of a function.
Inflexion point.
The second derivative. If a derivative f ' ( x ) of a function f ( x ) is differentiable
in the point ( x_{0} ), then its derivative is called the second derivative of the function
f ( x ) in the point ( x_{0} ) and denoted as f '' ( x_{0} ).
A function f ( x ) is called convex in an interval ( a, b ), if the function f ( x ) graph
is placed in this interval lower than a tangent line, going through any point ( x_{0}, f ( x_{0} ) ),
x_{0} ( a, b ).
A function f ( x ) is called concave in an interval ( a, b ), if the function f ( x ) graph
is placed in this interval higher than a tangent line, going through any point ( x_{0}, f ( x_{0} ) ),
x_{0} ( 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 x_{0}, then f '' ( x_{0} ) = 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}.

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