The second derivative, inflection point

The concept of the second derivative

Suppose the function has a derivative at all points of some interval. This derivative, in turn, is a function of If the function is differentiated, its derivative is called the second derivative and denoted (or )

Example.

The concept of convexity, concavity and points of inflection of the graph funct

Let the function defined on the interval and at the point has a finite derivative. Then schedule this function at the point can hold tangent

If in some neighborhood of the point all the points of the curve graph of a function (except for the points ) lie above the tangent line, then we say that the curve (function) at the point is convex (more precisely, strictly convex). Also it is sometimes said that in this case the function graph is convex down

If in some neighborhood of the point all the points of the curve (except the points ) lie below the tangent, then we say that the curve (function) at the point is potou (or rather, strictly potou). Also it is sometimes said that in this case the function graph is convex up

If the point is on the x-axis has the property that if the argument through the curve passes from one side to the other tangent, then the point is called the inflection point of the function point curve — point of inflection of the graph of a function

the point of inflection of the graph of a function

the inflection point of the function

In some neighborhood of the point : when the curve is below the tangent, and when the curve is above the tangent (or Vice versa)

The study of the function of the bulge, unott and inflection points

Example.

1. To find the scope and the intervals on which the function is continuous

Scope:

The function is continuous at every point of its domain of definition

2. Find the second derivative





3. Find an internal point of determining where or not there

there is in the entire scope

when

4. Mark the resulting point on the scope, find the sign of the second derivative and the behavior of the function on each interval, which splits the definition area

5. To record the desired outcome of the study (intervals of convexity and concavity and points of inflection)

In the interval and in the interval graph of a function convexity directed downwards and in the interval the graph of the function sent bump up

Inflection points: i (at these points changes the sign)