Function extremes
Definition 2
A point $x_0$ is called a point of maximum of the function $f(x)$ if there exists a neighborhood of this point such that for all $x$ from this neighborhood the inequality $f(x)\le f(x_0)$ is satisfied.
Definition 3
A point $x_0$ is called a maximum point of the function $f(x)$ if there exists a neighborhood of this point such that for all $x$ from this neighborhood the inequality $f(x)\ge f(x_0)$ is satisfied.
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 an extremum, one can formulate theorems on sufficient and necessary conditions his existence.
Theorem 2
Sufficient extremum condition
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 $f"(x)$ exist and retain a constant sign. Then:
1) If on the interval $(a,x_0)$ the derivative $f"\left(x\right)>0$, and on the interval $(x_0,b)$ the derivative $f"\left(x\right)
2) If the derivative $f"\left(x\right)0$ is on the interval $(a,x_0)$, 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 extremum points
The rule for examining a function for an extremum
2) Find the derivative $f"(x)$;
7) Draw conclusions about the presence of maxima and minima on each interval, using Theorem 2.
Function Ascending and Decreasing
Let us first introduce the definitions of increasing and decreasing functions.
Definition 5
A function $y=f(x)$ defined on an interval $X$ is called increasing if for any points $x_1,x_2\in X$ for $x_1
Definition 6
A function $y=f(x)$ defined on an interval $X$ is called decreasing if for any points $x_1,x_2\in X$ for $x_1f(x_2)$.
Examining a Function for Increasing and Decreasing
You can investigate functions for increasing and decreasing using the derivative.
In order to examine a function for intervals of increase and decrease, you must do the following:
1) Find the domain of the function $f(x)$;
2) Find the derivative $f"(x)$;
3) Find the points where the equality $f"\left(x\right)=0$;
4) Find points where $f"(x)$ does not exist;
5) Mark on the coordinate line all the found points and the domain of the given function;
6) Determine the sign of the derivative $f"(x)$ on each resulting interval;
7) Conclude: on the intervals where $f"\left(x\right)0$ the function increases.
Examples of problems for the study of functions for increasing, decreasing and the presence of extremum points
Example 1
Investigate the function for increasing and decreasing, and the presence of points of maxima and minima: $f(x)=(2x)^3-15x^2+36x+1$
Since the first 6 points are the same, we will draw 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:
\ \ if for any pair of points X and X", a ≤ x, the inequality f(x) ≤ f (x"), and strictly increasing - if the inequality f (x) f(x"). The decrease and strict decrease of a function are defined similarly. For example, the function at = X 2 (rice. , a) is strictly increasing on the segment , and
(rice. , b) strictly decreases on this interval. Increasing functions are denoted f (x), and decreasing f (x)↓. In order for a differentiable function f (x) was increasing on the interval [ a, b], it is necessary and sufficient that its derivative f"(x) was non-negative on [ a, b].
Along with the increase and decrease of a function on a segment, the increase and decrease of a function at a point are considered. Function at = f (x) is called increasing at the point x 0 if there is such an interval (α, β) containing the point x 0 , which for any point X from (α, β), x> x 0 , the inequality f (x 0) ≤ f (x), and for any point X from (α, β), x 0 , the inequality f (x) ≤ f (x 0). The strict increase of a function at a point is defined similarly x 0 . If a f"(x 0) > 0, then the function f(x) is strictly increasing at the point x 0 . If a f (x) increases at each point of the interval ( a, b), then it increases on this interval.
S. B. Stechkin.
Great Soviet Encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .
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