How to find out where a function increases and decreases. Sufficient signs of increasing and decreasing function

Extrema of the function

Definition 2

A point $x_0$ is called a maximum point of a function $f(x)$ if there is a neighborhood of this point such that for all $x$ in this neighborhood the inequality $f(x)\le f(x_0)$ holds.

Definition 3

A point $x_0$ is called a maximum point of a function $f(x)$ if there is a neighborhood of this point such that for all $x$ in this neighborhood the inequality $f(x)\ge f(x_0)$ holds.

The concept of an extremum of a function is closely related to the concept of a critical point of a function. Let us introduce its definition.

Definition 4

$x_0$ is called a critical point of the function $f(x)$ if:

1) $x_0$ - internal point of the domain of definition;

2) $f"\left(x_0\right)=0$ or does not exist.

For the concept of extremum, we can formulate theorems on sufficient and necessary conditions for its existence.

Theorem 2

Sufficient condition for an extremum

Let the point $x_0$ be critical for the function $y=f(x)$ and lie in the interval $(a,b)$. Let on each interval $\left(a,x_0\right)\ and\ (x_0,b)$ the derivative $f"(x)$ exists and maintains a constant sign. Then:

1) If on the interval $(a,x_0)$ the derivative is $f"\left(x\right)>0$, and on the interval $(x_0,b)$ the derivative is $f"\left(x\right)

2) If on the interval $(a,x_0)$ the derivative $f"\left(x\right)0$, then the point $x_0$ is the minimum point for this function.

3) If both on the interval $(a,x_0)$ and on the interval $(x_0,b)$ the derivative $f"\left(x\right) >0$ or the derivative $f"\left(x\right)

This theorem is illustrated in Figure 1.

Figure 1. Sufficient condition for the existence of extrema

Examples of extremes (Fig. 2).

Figure 2. Examples of extreme points

Rule for studying a function for extremum

2) Find the derivative $f"(x)$;

7) Draw conclusions about the presence of maxima and minima on each interval, using Theorem 2.

Increasing and decreasing function

Let us first introduce the definitions of increasing and decreasing functions.

Definition 5

A function $y=f(x)$ defined on the interval $X$ is said to be increasing if for any points $x_1,x_2\in X$ at $x_1

Definition 6

A function $y=f(x)$ defined on the interval $X$ is said to be decreasing if for any points $x_1,x_2\in X$ for $x_1f(x_2)$.

Studying a function for increasing and decreasing

You can study increasing and decreasing functions using the derivative.

In order to examine a function for intervals of increasing and decreasing, you must do the following:

1) Find the domain of definition of the function $f(x)$;

2) Find the derivative $f"(x)$;

3) Find the points at which the equality $f"\left(x\right)=0$ holds;

4) Find the points at which $f"(x)$ does not exist;

5) Mark on the coordinate line all the points found and the domain of definition of this function;

6) Determine the sign of the derivative $f"(x)$ on each resulting interval;

7) Draw a conclusion: on intervals where $f"\left(x\right)0$ the function increases.

Examples of problems for studying functions for increasing, decreasing and the presence of extrema points

Example 1

Examine the function for increasing and decreasing, and the presence of maximum and minimum points: $f(x)=(2x)^3-15x^2+36x+1$

Since the first 6 points are the same, let’s go through them first.

1) Domain of definition - all real numbers;

2) $f"\left(x\right)=6x^2-30x+36$;

3) $f"\left(x\right)=0$;

\ \ \

4) $f"(x)$ exists at all points of the domain of definition;

5) Coordinate line:

Figure 3.

6) Determine the sign of the derivative $f"(x)$ on each interval:

\ \ .

Sufficient conditions for the extremum of a function.

To find the maxima and minima of a function, you can use any of the three signs of extremum, of course, if the function satisfies their conditions. The most common and convenient is the first of them.

The first sufficient condition for an extremum.

Let the function y=f(x) be differentiable in the -neighborhood of the point and continuous at the point itself.

In other words:

Algorithm for finding extremum points based on the first sign of extremum of a function.

We find the domain of definition of the function.
We find the derivative of the function on the domain of definition.
We determine the zeros of the numerator, the zeros of the denominator of the derivative and the points of the domain of definition in which the derivative does not exist (all listed points are called points of possible extremum, passing through these points, the derivative can just change its sign).
These points divide the domain of definition of the function into intervals in which the derivative retains its sign. We determine the signs of the derivative on each of the intervals (for example, by calculating the value of the derivative of a function at any point in a particular interval).
We select points at which the function is continuous and, passing through which, the derivative changes sign - these are the extremum points.

There are too many words, let’s better look at a few examples of finding extremum points and extrema of a function using the first sufficient condition for the extremum of a function.

Example.

Find the extrema of the function.

Solution.

The domain of a function is the entire set of real numbers except x=2.

Finding the derivative:

The zeros of the numerator are the points x=-1 and x=5, the denominator goes to zero at x=2. Mark these points on the number axis

We determine the signs of the derivative at each interval; to do this, we calculate the value of the derivative at any of the points of each interval, for example, at the points x=-2, x=0, x=3 and x=6.

Therefore, on the interval the derivative is positive (in the figure we put a plus sign over this interval). Likewise

Therefore, we put a minus above the second interval, a minus above the third, and a plus above the fourth.

It remains to select points at which the function is continuous and its derivative changes sign. These are the extremum points.

At the point x=-1 the function is continuous and the derivative changes sign from plus to minus, therefore, according to the first sign of extremum, x=-1 is the maximum point, the maximum of the function corresponds to it .

At the point x=5 the function is continuous and the derivative changes sign from minus to plus, therefore, x=-1 is the minimum point, the minimum of the function corresponds to it .

Graphic illustration.

Answer:

PLEASE NOTE: the first sufficient criterion for an extremum does not require differentiability of the function at the point itself.

Example.

Find extremum points and extrema of the function .

Solution.

The domain of a function is the entire set of real numbers. The function itself can be written as:

Let's find the derivative of the function:

At the point x=0 the derivative does not exist, since the values of the one-sided limits do not coincide when the argument tends to zero:

At the same time, the original function is continuous at the point x=0 (see the section on studying the function for continuity):

Let's find the value of the argument at which the derivative goes to zero:

Let's mark all the obtained points on the number line and determine the sign of the derivative on each of the intervals. To do this, we calculate the values of the derivative at arbitrary points of each interval, for example, at x=-6, x=-4, x=-1, x=1, x=4, x=6.