Program of Lessons

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 ( 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 the function f ( x ) graph 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 the function f ( x ) graph 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.

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