Whole rational function

An entire rational function (also a polynomial function ) is a numerical function defined by a polynomial . The simplest representatives of an entire rational function are constant , linear, and quadratic functions.

Along with fractional rational functions , entire rational functions are a special case of rational functions .

Content

1 Definition
- 1.1 Notes
2 Types
- 2.1 Examples
3 Basic properties
- 3.1 Scope, multiple values, limits
- 3.2 Parity and symmetry
- 3.3 Derivative and antiderivative
4 Singular points of a polynomial function
- 4.1 Calculation of function zeros
- 4.2 Monotonicity and extremum points
- 4.3 Convexity and inflection points
- 4.4 Graphic relationship between singular points
5 Literature
6 References

Definition

An entire rational function is a function of one real variable of the form:

f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb +a_{2}x^{2}+a_{1}x+a_{0}=\sum _{i=0}^{n}a_{i}x^{i},

{\ displaystyle f (x) = a_ {n} x ^ {n} + a_ {n-1} x ^ {n-1} + \ dotsb + a_ {2} x ^ {2} + a_ {1} x + a_ {0} = \ sum _ {i = 0} ^ {n} a_ {i} x ^ {i},}

f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb +a_{2}x^{2}+a_{1}x+a_{0}=\sum _{i=0}^{n}a_{i}x^{i},

Where $n\in \mathbb {N}$ ${\ displaystyle n \ in \ mathbb {N}}$ $n\in \mathbb{N}$ , $a_{n},a_{n-1},\ldots ,a_{2},a_{1},a_{0}\in \mathbb {R}$ ${\ displaystyle a_ {n}, a_ {n-1}, \ ldots, a_ {2}, a_ {1}, a_ {0} \ in \ mathbb {R}}$ $a_{n},a_{n-1},\ldots ,a_{2},a_{1},a_{0}\in \mathbb {R}$ and $a_{n}\neq 0$ ${\ displaystyle a_ {n} \ neq 0}$ $a_{n}\neq 0$ .

In other words, an entire rational function is a linear combination of several power functions .

Remarks

A special case of an entire rational function is a function $f(x)=0$ ${\ displaystyle f (x) = 0}$ $f(x)=0$ , all coefficients of which are equal to zero.

Natural number $n$ ${\ displaystyle n}$ $n$ (the highest degree of variable $x$ ${\ displaystyle x}$ $x$ ) determines the degree of the polynomial function.
Real numbers $a_{n},\ldots ,a_{2},a_{1},a_{0}$ ${\ displaystyle a_ {n}, \ ldots, a_ {2}, a_ {1}, a_ {0}}$ $a_{n},\ldots ,a_{2},a_{1},a_{0}$ are called the coefficients of the polynomial function. Moreover, the number $a_{n}$ ${\ displaystyle a_ {n}}$ $a_n$ often called the highest coefficient, and the number $a_{0}$ ${\ displaystyle a_ {0}}$ $a_{0}$ - free ratio.

n=0

{\ displaystyle n = 0}

n=0

n=1

{\ displaystyle n = 1}

n=1

n=2

{\ displaystyle n = 2}

n=2

n=3

{\ displaystyle n = 3}

n=4

{\ displaystyle n = 4}

n=5

{\ displaystyle n = 5}

Types

At $n=0$ ${\ displaystyle n = 0}$ $n=0$ polynomial function degenerates into a constant function $f(x)=a_{0}$ ${\ displaystyle f (x) = a_ {0}}$
At $n=1$ ${\ displaystyle n = 1}$ $n = 1$ we get a linear function $f(x)=a_{1}x+a_{0}$ ${\ displaystyle f (x) = a_ {1} x + a_ {0}}$ .
At $n=2$ ${\ displaystyle n = 2}$ we get a quadratic function $f(x)=a_{2}x^{2}+a_{1}x+a_{0}$ ${\ displaystyle f (x) = a_ {2} x ^ {2} + a_ {1} x + a_ {0}}$ .
At $n=3$ ${\ displaystyle n = 3}$ it turns out a cubic function $f(x)=a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}$ ${\ displaystyle f (x) = a_ {3} x ^ {3} + a_ {2} x ^ {2} + a_ {1} x + a_ {0}}$ .
If $a_{n}\neq 0$ ${\ displaystyle a_ {n} \ neq 0}$ and all other coefficients are equal $0$ ${\ displaystyle 0}$ , a power function takes place $f(x)=a_{n}x^{n}$ ${\ displaystyle f (x) = a_ {n} x ^ {n}}$ with a natural indicator.

Examples

Function $f(x)=-2x^{3}+3x^{2}-5x+4$ ${\ displaystyle f (x) = - 2x ^ {3} + 3x ^ {2} -5x + 4}$ is a polynomial function of the third degree with coefficients $-2$ ${\ displaystyle -2}$ ; $3$ ${\ displaystyle 3}$ ; $-5$ ${\ displaystyle -5}$ and $four$ ${\ displaystyle 4}$ .
Function $f(x)=x^{5}+3x^{3}+5x$ ${\ displaystyle f (x) = x ^ {5} + 3x ^ {3} + 5x}$ is a polynomial function of the fifth degree with coefficients $one$ ${\ displaystyle 1}$ ; $0$ ${\ displaystyle 0}$ ; $3$ ${\ displaystyle 3}$ ; $0$ ${\ displaystyle 0}$ ; $5$ ${\ displaystyle 5}$ and $0$ ${\ displaystyle 0}$ .
Function $f(x)={\frac {1}{3}}x^{2}-{\sqrt {2}}$ ${\ displaystyle f (x) = {\ frac {1} {3}} x ^ {2} - {\ sqrt {2}}}$ is a polynomial function of the second degree (i.e., a quadratic function) with coefficients ${\frac {1}{3}}$ ${\ displaystyle {\ frac {1} {3}}}$ and $-{\sqrt {2}}$ ${\ displaystyle - {\ sqrt {2}}}$ .

Key Features

Scope, set of values, limits

A polynomial function over a field of real numbers is defined everywhere and is continuous over its entire domain of definition. Its set of values is also a subset of the set of real numbers. With even $n$ ${\ displaystyle n}$ many values will be, depending on the sign of the highest coefficient $a_{n}$ ${\ displaystyle a_ {n}}$ , bounded above or below (see also table).

The limit of a polynomial function at infinity $x\to \pm \infty$ ${\ displaystyle x \ to \ pm \ infty}$ always exists, and its specific meaning depends on the parity of the degree $n$ ${\ displaystyle n}$ and sign at the highest coefficient $a_{n}$ ${\ displaystyle a_ {n}}$ . In this case, the graph of the polynomial function behaves exactly the same as the graph of the power function $g(x)=a_{n}x^{n}$ ${\ displaystyle g (x) = a_ {n} x ^ {n}}$ :

	even $n$ ${\ displaystyle n}$		odd $n$ ${\ displaystyle n}$
$a_{n}>0$ ${\ displaystyle a_ {n}> 0}$	$f(x)\to +\infty$ ${\ displaystyle f (x) \ to + \ infty}$ at $x\to \pm \infty$ ${\ displaystyle x \ to \ pm \ infty}$ (many values are bounded below)		$f(x)\to -\infty$ ${\ displaystyle f (x) \ to - \ infty}$ at $x\to -\infty$ ${\ displaystyle x \ to - \ infty}$ $f(x)\to +\infty$ ${\ displaystyle f (x) \ to + \ infty}$ at $x\to +\infty$ ${\ displaystyle x \ to + \ infty}$
$a_{n}<0$ ${\ displaystyle a_ {n} <0}$	$f(x)\to -\infty$ ${\ displaystyle f (x) \ to - \ infty}$ at $x\to \pm \infty$ ${\ displaystyle x \ to \ pm \ infty}$ (many values are bounded above)		$f(x)\to +\infty$ ${\ displaystyle f (x) \ to + \ infty}$ at $x\to -\infty$ ${\ displaystyle x \ to - \ infty}$ $f(x)\to -\infty$ ${\ displaystyle f (x) \ to - \ infty}$ at $x\to +\infty$ ${\ displaystyle x \ to + \ infty}$

The limit of the polynomial function at each point $x_{0}$ ${\ displaystyle x_ {0}}$ coincides with the value of the function at this point: $\lim _{x\to x_{0}}f(x)=f(x_{0})$ ${\ displaystyle \ lim _ {x \ to x_ {0}} f (x) = f (x_ {0})}$ .

For example, for a function $f(x)=3x^{8}-5x^{4}+1$ ${\ displaystyle f (x) = 3x ^ {8} -5x ^ {4} +1}$ we have:

\lim _{x\to -\infty }f(x)=\lim _{x\to \infty }(3x^{8})=+\infty

{\ displaystyle \ lim _ {x \ to - \ infty} f (x) = \ lim _ {x \ to \ infty} (3x ^ {8}) = + \ infty}

\lim _{x\to 1}f(x)=f(1)=3-5+1=-1

{\ displaystyle \ lim _ {x \ to 1} f (x) = f (1) = 3-5 + 1 = -1}

Parity and Symmetry

Function

f(x)=x^{4}-4x^{2}+3

{\ displaystyle f (x) = x ^ {4} -4x ^ {2} +3}

contains only even exponents (

four

{\ displaystyle 4}

;

2

{\ displaystyle 2}

and

0

{\ displaystyle 0}

), therefore, its graph has axial symmetry with respect to the ordinate axis

A polynomial function is even if all exponents in its record are even numbers . The graph of such a function has axial symmetry with respect to the ordinate axis ). This symmetry holds in view of the equality $f(-x)=f(x)$ ${\ displaystyle f (-x) = f (x)}$ fair to even functions. The following polynomial functions are even, for example:

$f(x)=-2x^{4}+3x^{2}$ ${\ displaystyle f (x) = - 2x ^ {4} + 3x ^ {2}}$ with indicators $four$ ${\ displaystyle 4}$ and $2$ ${\ displaystyle 2}$
$f(x)=-2x^{4}+3x^{2}+1$ ${\ displaystyle f (x) = - 2x ^ {4} + 3x ^ {2} +1}$ with indicators $four$ ${\ displaystyle 4}$ ; $2$ ${\ displaystyle 2}$ and $0$ ${\ displaystyle 0}$
$f(x)=5x^{6}-2$ ${\ displaystyle f (x) = 5x ^ {6} -2}$ with indicators $6$ ${\ displaystyle 6}$ and $0$ ${\ displaystyle 0}$

A polynomial function is odd if all exponents in its record are odd numbers. The graph of such a function has central symmetry with respect to the center of the coordinate system ). This symmetry holds in view of the equality $f(-x)=-f(x)$ ${\ displaystyle f (-x) = - f (x)}$ performed for odd functions. Odd are, for example, the following polynomial functions:

$f(x)=2x^{7}-3x^{5}$ ${\ displaystyle f (x) = 2x ^ {7} -3x ^ {5}}$ with indicators $7$ ${\ displaystyle 7}$ and $5$ ${\ displaystyle 5}$
$f(x)=-2x^{5}+3x^{3}-2x$ ${\ displaystyle f (x) = - 2x ^ {5} + 3x ^ {3} -2x}$ with indicators $5$ ${\ displaystyle 5}$ ; $3$ ${\ displaystyle 3}$ and $one$ ${\ displaystyle 1}$

If the record of a polynomial function contains both even and odd exponents, such a function is neither even nor even . For this reason, its graph does not have symmetry either with respect to the ordinate axis or with respect to the center of the coordinate system. However, such functions may have more complicated cases of symmetry. In particular, the following statements are true:

If $f(x_{0}+x)=f(x_{0}-x)$ ${\ displaystyle f (x_ {0} + x) = f (x_ {0} -x)}$ for some number $x_{0}\in \mathbb {R}$ ${\ displaystyle x_ {0} \ in \ mathbb {R}}$ , then the graph of this function has axial symmetry with respect to the line $x=x_{0}$ ${\ displaystyle x = x_ {0}}$ .
If $f(x_{0}+x)-y_{0}=-f(x_{0}-x)+y_{0}$ ${\ displaystyle f (x_ {0} + x) -y_ {0} = - f (x_ {0} -x) + y_ {0}}$ for some pair of numbers $x_{0},y_{0}\in \mathbb {R}$ ${\ displaystyle x_ {0}, y_ {0} \ in \ mathbb {R}}$ , then the graph of this function has central symmetry with respect to the point $P(x_{0}|y_{0})$ ${\ displaystyle P (x_ {0} | y_ {0})}$ .

In addition, the following properties also apply:

The graph of each polynomial function of the second degree is symmetric with respect to a line passing parallel to the ordinate axis through the vertex of the parabola , which is also the extremum point of this function.
The graph of each polynomial function of the third degree is symmetric with respect to its inflection point .

Derivative and antiderivative

Differentiation rules

$(u(x)+v(x))'=u'(x)+v'(x)$ ${\ displaystyle (u (x) + v (x)) '= u' (x) + v '(x)}$
$(a\cdot x^{n})'=a\cdot n\cdot x^{n-1}$ ${\ displaystyle (a \ cdot x ^ {n}) '= a \ cdot n \ cdot x ^ {n-1}}$
$(a)'=0$ ${\ displaystyle (a) '= 0}$

Integration Rules

$\int [u(x)+v(x)]dx=\int u(x)dx+\int v(x)dx$ ${\ displaystyle \ int [u (x) + v (x)] dx = \ int u (x) dx + \ int v (x) dx}$
$\int (a\cdot x^{n})dx={\dfrac {a}{n+1}}\cdot x^{n+1}+C$ ${\ displaystyle \ int (a \ cdot x ^ {n}) dx = {\ dfrac {a} {n + 1}} \ cdot x ^ {n + 1} + C}$
$\int \!a\,dx=ax+C$ ${\ displaystyle \ int \! a \, dx = ax + C}$

A polynomial function is differentiable in its entire domain . Its derivative is easily found using elementary rules of differentiation. So, the derivative of the function $f(x)=2x^{3}-4x^{2}+5x-1$ ${\ displaystyle f (x) = 2x ^ {3} -4x ^ {2} + 5x-1}$ calculated as follows:

{\begin{aligned}f'(x)&=(2x^{3}-4x^{2}+5x-1)'=\\&=(2x^{3})'+(-4x^{2})'+(5x)'+(-1)'=\\&=2\cdot 3x^{2}-4\cdot 2x+5\cdot 1x^{0}-1\cdot 0=\\&=6x^{2}-8x+5=\end{aligned}}

{\ displaystyle {\ begin {aligned} f '(x) & = (2x ^ {3} -4x ^ {2} + 5x-1)' = \\ & = (2x ^ {3}) '+ (- 4x ^ {2}) '+ (5x)' + (- 1) '= \\ & = 2 \ cdot 3x ^ {2} -4 \ cdot 2x + 5 \ cdot 1x ^ {0} -1 \ cdot 0 = \\ & = 6x ^ {2} -8x + 5 = \ end {aligned}}}

A polynomial function is also integrable in its entire domain of definition . Its antiderivative is also easily found with the help of elementary integration rules. For example, the antiderivative of the same function $f(x)$ ${\ displaystyle f (x)}$ , as in the example above, is calculated as follows:

F(x)=\int (2x^{3}-4x^{2}+5x-1)dx={\frac {1}{2}}x^{4}-{\frac {4}{3}}x^{3}+{\frac {5}{2}}x^{2}-x+c

{\ displaystyle F (x) = \ int (2x ^ {3} -4x ^ {2} + 5x-1) dx = {\ frac {1} {2}} x ^ {4} - {\ frac {4 } {3}} x ^ {3} + {\ frac {5} {2}} x ^ {2} -x + c}

where

c\in \mathbb {R} .

{\ displaystyle c \ in \ mathbb {R}.}

It is easy to see that the derivative and the antiderivative of the polynomial function $f(x)$ ${\ displaystyle f (x)}$ degrees of $n$ ${\ displaystyle n}$ also themselves are polynomial. In this case, the function $f'(x)$ ${\ displaystyle f '(x)}$ has a degree $n-1$ ${\ displaystyle n-1}$ and function $F(x)$ ${\ displaystyle F (x)}$ - degree $n+1$ ${\ displaystyle n + 1}$ (except for the trivial case when $f(x)=0$ ${\ displaystyle f (x) = 0}$ )

Singular points of a polynomial function

Calculating function zeros

The zeros of the polynomial function coincide with the roots of the polynomial present in its equation. Thus, to find zeros, it is necessary to solve the equation $f(x)=0$ ${\ displaystyle f (x) = 0}$ . The solution method largely depends on the particular equation of the function.

If the function $f(x)=a_{n}x^{n}+\dotsb +a_{2}x^{2}+a_{1}x+a_{0}$ ${\ displaystyle f (x) = a_ {n} x ^ {n} + \ dotsb + a_ {2} x ^ {2} + a_ {1} x + a_ {0}}$ recorded in factorized form $f(x)=a_{n}\cdot (x-x_{1})^{k_{1}}\cdot (x-x_{2})^{k_{2}}\dotsb (x-x_{m})^{k_{m}}$ ${\ displaystyle f (x) = a_ {n} \ cdot (x-x_ {1}) ^ {k_ {1}} \ cdot (x-x_ {2}) ^ {k_ {2}} \ dotsb (x -x_ {m}) ^ {k_ {m}}}$ , where each of the factors is a linear binomial , then real numbers $x_{1}$ ${\ displaystyle x_ {1}}$ , $x_{2}$ ${\ displaystyle x_ {2}}$ , ..., $x_{m}$ ${\ displaystyle x_ {m}}$ are zeros of function $f(x)$ ${\ displaystyle f (x)}$ , and natural numbers $k_{1}$ ${\ displaystyle k_ {1}}$ , $k_{2}$ ${\ displaystyle k_ {2}}$ , ..., $k_{m}$ ${\ displaystyle k_ {m}}$ show the multiplicity of the corresponding zeros of this function. In this case, the condition is satisfied: $k_{1}+k_{2}+\dotsb +k_{m}\leq n$ ${\ displaystyle k_ {1} + k_ {2} + \ dotsb + k_ {m} \ leq n}$ . Thus degree $n$ ${\ displaystyle n}$ the functions $f(x)$ ${\ displaystyle f (x)}$ determines the maximum possible number of its zeros over the field of real numbers . In the case of a generalization of a polynomial function on the field of complex numbers , in accordance with the basic theorem of algebra , the following equality holds: $k_{1}+k_{2}+\dotsb +k_{m}=n$ ${\ displaystyle k_ {1} + k_ {2} + \ dotsb + k_ {m} = n}$ .

So, for example, a polynomial function $f(x)=-0{,}01\cdot x^{3}\cdot (x-2)\cdot (x+3)^{2}\cdot (x^{2}+1)$ ${\ displaystyle f (x) = - 0 {,} 01 \ cdot x ^ {3} \ cdot (x-2) \ cdot (x + 3) ^ {2} \ cdot (x ^ {2} +1) }$ has three zeros, namely: $x_{1}=0$ ${\ displaystyle x_ {1} = 0}$ (multiplicity 3), $x_{2}=2$ ${\ displaystyle x_ {2} = 2}$ (multiplicity 1) and $x_{3}=-3$ ${\ displaystyle x_ {3} = - 3}$ (multiplicity 2). Square binomial $x^{2}+1$ ${\ displaystyle x ^ {2} +1}$ it has no real roots, therefore it cannot be further factorized into linear factors.

In general, to find the zeros of a polynomial function of degree $n=1$ ${\ displaystyle n = 1}$ and $n=2$ ${\ displaystyle n = 2}$ The methods used to solve linear and quadratic equations, respectively, are used. To find the zeros of a polynomial degree function $n\geq 3$ ${\ displaystyle n \ geq 3}$ where possible, various special methods for solving algebraic equations of higher degrees can be used (in particular, this applies to biquadratic and power equations). In more general cases, either universal methods such as dividing polynomials by column or Horner scheme are used , which allow, however, to find only integer (exact) solutions, or numerical methods (for example, Newton's method ) are used to find all (but only approximate) solutions.

The methods for finding the integer roots of a polynomial are based on a corollary of Bezout's theorem . In particular, to factorize a polynomial function $f(x)=a_{n}x^{n}+\dotsb +a_{2}x^{2}+a_{1}x+a_{0}$ ${\ displaystyle f (x) = a_ {n} x ^ {n} + \ dotsb + a_ {2} x ^ {2} + a_ {1} x + a_ {0}}$ with integer coefficients first among all the divisors of the free coefficient $a_{0}$ ${\ displaystyle a_ {0}}$ any one root is selected $x_{0}$ ${\ displaystyle x_ {0}}$ , that is, such an integer for which it is true: $f(x_{0})=0$ ${\ displaystyle f (x_ {0}) = 0}$ . Then, by dividing by a column or using the Horner scheme, $f(x)$ ${\ displaystyle f (x)}$ on the binomial $x-x_{0}$ ${\ displaystyle x-x_ {0}}$ factorization of the original polynomial to the form $f(x)=(x-x_{0})\cdot g(x)$ ${\ displaystyle f (x) = (x-x_ {0}) \ cdot g (x)}$ where $g(x)$ ${\ displaystyle g (x)}$ - degree polynomial $n-1$ ${\ displaystyle n-1}$ . Thus, the degree of the original function, and hence its complexity, decreases. Finding function zeros $f(x)$ ${\ displaystyle f (x)}$ comes down to finding zeros of a function $g(x)$ ${\ displaystyle g (x)}$ .

So, for example, to find the zeros of a function $f(x)=x^{3}-12x^{2}+5x+150$ ${\ displaystyle f (x) = x ^ {3} -12x ^ {2} + 5x + 150}$ (see example) with integer coefficients, one root is first “guessed” (number $5$ ${\ displaystyle 5}$ is among the divisors of the number $150$ ${\ displaystyle 150}$ ) and then the original polynomial $f(x)$ ${\ displaystyle f (x)}$ divided by a binomial $x-5$ ${\ displaystyle x-5}$ . Further finding the remaining zeros of the function $f(x)$ ${\ displaystyle f (x)}$ comes down to finding the zeros of the resulting function $g(x)=x^{2}-7x-30$ ${\ displaystyle g (x) = x ^ {2} -7x-30}$ which can be easily found by solving the corresponding quadratic equation.

Monotonicity and extreme points

Both functions

f(x)=x^{3}

{\ displaystyle f (x) = x ^ {3}}

and

g(x)=x^{2}

{\ displaystyle g (x) = x ^ {2}}

have zero first derivative at

x_{0}=0

{\ displaystyle x_ {0} = 0}

. However

f''(0)=0

{\ displaystyle f '' (0) = 0}

and

g''(0)=2

{\ displaystyle g '' (0) = 2}

. If for

g(x)

{\ displaystyle g (x)}

this means that there is a local minimum in

x_{0}

{\ displaystyle x_ {0}}

then for

f(x)

{\ displaystyle f (x)}

no conclusion can be drawn from the second derivative.

Since a necessary condition for the existence of a local extremum of a function at a point $x_{0}$ ${\ displaystyle x_ {0}}$ is the zero value of the angular coefficient in it, then to find the extrema of the polynomial function it is necessary to solve the equation $f'(x)=0$ ${\ displaystyle f '(x) = 0}$ , that is, calculate the zeros of its derivative function. Since the derivative of the polynomial function itself is a polynomial function (of a lower degree), to find potential points of extremum $x_{0}$ ${\ displaystyle x_ {0}}$ the same methods are used as for calculating the zeros of the function itself. From the property on the number of roots of a polynomial we can conclude that a polynomial function of degree $n$ ${\ displaystyle n}$ could theoretically have up $n-1$ ${\ displaystyle n-1}$ local extremes. It is also easy to see that between any two zeros of the polynomial function, at least one local extremum is necessarily located.

Since any polynomial function $f(x)$ ${\ displaystyle f (x)}$ continuous and twice differentiable at each point $x_{0}$ ${\ displaystyle x_ {0}}$ , then to verify the existence of a local maximum and a local minimum of the polynomial function, it is enough to verify that the found value $x_{0}$ ${\ displaystyle x_ {0}}$ (zero derivative of the function) satisfies one of the sufficient criteria.

Criterion for the second derivative:
If $f'(x_{0})=0$ ${\ displaystyle f '(x_ {0}) = 0}$ and $f''(x_{0})<0$ ${\ displaystyle f '' (x_ {0}) <0}$ then $x_{0}$ ${\ displaystyle x_ {0}}$ is the point of local maximum.
If $f'(x_{0})=0$ ${\ displaystyle f '(x_ {0}) = 0}$ and $f''(x_{0})>0$ ${\ displaystyle f '' (x_ {0})> 0}$ then $x_{0}$ ${\ displaystyle x_ {0}}$ is a local minimum point.
If $f'(x_{0})=0$ ${\ displaystyle f '(x_ {0}) = 0}$ and $f''(x_{0})=0$ ${\ displaystyle f '' (x_ {0}) = 0}$ then about the point $x_{0}$ ${\ displaystyle x_ {0}}$ no conclusion can be drawn.
Criterion for the first derivative:
If $f'(x_{0})=0$ ${\ displaystyle f '(x_ {0}) = 0}$ and $f'(x)$ ${\ displaystyle f '(x)}$ changes sign from plus to minus when crossing a point $x_{0}$ ${\ displaystyle x_ {0}}$ then $x_{0}$ ${\ displaystyle x_ {0}}$ is the point of local maximum.
If $f'(x_{0})=0$ ${\ displaystyle f '(x_ {0}) = 0}$ and $f'(x)$ ${\ displaystyle f '(x)}$ changes sign from minus to plus when crossing a point $x_{0}$ ${\ displaystyle x_ {0}}$ then $x_{0}$ ${\ displaystyle x_ {0}}$ is a local minimum point.
If $f'(x_{0})=0$ ${\ displaystyle f '(x_ {0}) = 0}$ and $f'(x)$ ${\ displaystyle f '(x)}$ does not change sign when crossing a point $x_{0}$ ${\ displaystyle x_ {0}}$ then $x_{0}$ ${\ displaystyle x_ {0}}$ is not a local minimum point (“ saddle point ”).

Bulge and inflection points

Function graph

f(x)=x^{3}+2x^{2}

{\ displaystyle f (x) = x ^ {3} + 2x ^ {2}}

changes at a point

W

{\ displaystyle W}

its bulge and therefore is located on different sides of the tangent drawn at this point

Function

g(x)=x^{4}-x

{\ displaystyle g (x) = x ^ {4} -x}

has no inflection point in

x_{0}=0

{\ displaystyle x_ {0} = 0}

although it holds

g''(0)=0

{\ displaystyle g '' (0) = 0}

, therefore, its graph is completely located on one side of the tangent at this point

A necessary condition for the existence of an inflection point of a function at $x_{0}$ ${\ displaystyle x_ {0}}$ (that is, the point at which the convexity of the graph of the function changes) is the zero value of the second derivative in it. Thus, to find the inflection points of a polynomial function, it is necessary to solve the equation $f''(x)=0$ ${\ displaystyle f '' (x) = 0}$ . From the property on the number of roots of a polynomial we can conclude that a polynomial function of degree $n$ ${\ displaystyle n}$ may have up $n-2$ ${\ displaystyle n-2}$ inflection points.

In view of the continuity and multiple differentiability of the polynomial function $f(x)$ ${\ displaystyle f (x)}$ at every point $x_{0}$ ${\ displaystyle x_ {0}}$ to check for the existence of inflection points, it is enough to verify that the value found $x_{0}$ ${\ displaystyle x_ {0}}$ (zero of the second derivative) satisfies one of the sufficient criteria.

Criterion for the third derivative:
If $f''(x_{0})=0$ ${\ displaystyle f '' (x_ {0}) = 0}$ and $f'''(x_{0})\neq 0$ ${\ displaystyle f '' '(x_ {0}) \ neq 0}$ then point $x_{0}$ ${\ displaystyle x_ {0}}$ is the inflection point.
If $f''(x_{0})=0$ ${\ displaystyle f '' (x_ {0}) = 0}$ and $f'''(x_{0})=0$ ${\ displaystyle f '' '(x_ {0}) = 0}$ then about the point $x_{0}$ ${\ displaystyle x_ {0}}$ no conclusion can be drawn.
Criterion for the second derivative:
If $f''(x_{0})=0$ ${\ displaystyle f '' (x_ {0}) = 0}$ and $f''(x)$ ${\ displaystyle f '' (x)}$ changes sign when crossing a point $x_{0}$ ${\ displaystyle x_ {0}}$ then $x_{0}$ ${\ displaystyle x_ {0}}$ is the inflection point.
If $f''(x_{0})=0$ ${\ displaystyle f '' (x_ {0}) = 0}$ and $f''(x)$ ${\ displaystyle f '' (x)}$ does not change sign when crossing a point $x_{0}$ ${\ displaystyle x_ {0}}$ then $x_{0}$ ${\ displaystyle x_ {0}}$ not an inflection point.

For example, to find the inflection points of a function $f(x)=x^{3}+2x^{2}$ ${\ displaystyle f (x) = x ^ {3} + 2x ^ {2}}$ The following calculations are performed:

f'(x)=3x^{2}+4x,\quad f''(x)=6x+4,\quad f'''(x)=6.

{\ displaystyle f '(x) = 3x ^ {2} + 4x, \ quad f' '(x) = 6x + 4, \ quad f' '' (x) = 6.}

As $f''(x)=0$ ${\ displaystyle f '' (x) = 0}$ at $x=-{\frac {2}{3}}$ ${\ displaystyle x = - {\ frac {2} {3}}}$ and $f'''(-{\frac {2}{3}})=6\neq 0$ ${\ displaystyle f '' '(- {\ frac {2} {3}}) = 6 \ neq 0}$ then in $x_{0}=-{\frac {2}{3}}$ ${\ displaystyle x_ {0} = - {\ frac {2} {3}}}$ there is an inflection point.

Function at the same time $g(x)=x^{4}-x$ ${\ displaystyle g (x) = x ^ {4} -x}$ has no inflection point in $x_{0}=0$ ${\ displaystyle x_ {0} = 0}$ , despite the fact that the conditions are met:

g'(x)=4x^{3}-1,\quad f''(x)=12x^{2},\quad f'''(x)=24x.

{\ displaystyle g '(x) = 4x ^ {3} -1, \ quad f' '(x) = 12x ^ {2}, \ quad f' '' (x) = 24x.}

As $g''(x)=0$ ${\ displaystyle g '' (x) = 0}$ at $x_{0}=0$ ${\ displaystyle x_ {0} = 0}$ but $g'''(0)=0$ ${\ displaystyle g '' '(0) = 0}$ then it is necessary to use the criterion for the second derivative. Due to the fact that the function $g''(x)=12x^{2}$ ${\ displaystyle g '' (x) = 12x ^ {2}}$ can only take positive values, there is no sign change, therefore the function $g(x)$ ${\ displaystyle g (x)}$ has no inflection point in $x_{0}=0$ ${\ displaystyle x_ {0} = 0}$ .

Graphic Relationship between Feature Points

Zeros of function of different multiplicity

To determine the multiplicity of zeros of a polynomial function, the fact that any polynomial function is many times differentiable can be used. So if $x_{0}$ ${\ displaystyle x_ {0}}$ - zero multiplicity $k$ ${\ displaystyle k}$ (but not multiplicities $k+1$ ${\ displaystyle k + 1}$ ) polynomial function $f(x)$ ${\ displaystyle f (x)}$ then the following conditions are true:

f(x_{0})=0,\quad f'(x_{0})=0,\quad f''(x_{0})=0,\quad \dotsc ,\quad f^{(k-1)}(x_{0})=0,\quad f^{(k)}(x_{0})\neq 0.

{\ displaystyle f (x_ {0}) = 0, \ quad f '(x_ {0}) = 0, \ quad f' '(x_ {0}) = 0, \ quad \ dotsc, \ quad f ^ { (k-1)} (x_ {0}) = 0, \ quad f ^ {(k)} (x_ {0}) \ neq 0.}

For example, for a function $f(x)=x^{3}-3x^{2}+3x-1$ ${\ displaystyle f (x) = x ^ {3} -3x ^ {2} + 3x-1}$ fair: $f'(x)=3x^{2}-6x+3$ ${\ displaystyle f '(x) = 3x ^ {2} -6x + 3}$ ; $f''(x)=6x-6$ ${\ displaystyle f '' (x) = 6x-6}$ and $f'''(x)=6$ ${\ displaystyle f '' '(x) = 6}$ . As $f(1)=1-3+3-1=0$ ${\ displaystyle f (1) = 1-3 + 3-1 = 0}$ then $x_{0}=1$ ${\ displaystyle x_ {0} = 1}$ is a function zero $f(x)$ ${\ displaystyle f (x)}$ . The following is done: $f'(1)=3-6+3=0$ ${\ displaystyle f '(1) = 3-6 + 3 = 0}$ , $f''(1)=6-6=0$ ${\ displaystyle f '' (1) = 6-6 = 0}$ and $f'''(1)=6\neq 0$ ${\ displaystyle f '' '(1) = 6 \ neq 0}$ . In this way, $x_{0}=1$ ${\ displaystyle x_ {0} = 1}$ is a zero of multiplicity 3!

The multiplicity of zeros can be seen from the graph of the multinimal function:

In case of zero $x_{0}$ ${\ displaystyle x_ {0}}$ of multiplicity 1, the function graph intersects the abscissa axis . Moreover, at the point $x_{0}$ ${\ displaystyle x_ {0}}$ there is no change in the monotonicity of the function , since the second derivative at this point is not equal to zero!
If zero $x_{0}$ ${\ displaystyle x_ {0}}$ has an even multiplicity of 2, 4, 6, etc., then it is obvious that $f'(x_{0})=0$ ${\ displaystyle f '(x_ {0}) = 0}$ and $f''(x_{0})\neq 0$ ${\ displaystyle f '' (x_ {0}) \ neq 0}$ . Thus, the function graph will touch the abscissa at the point $x_{0}$ ${\ displaystyle x_ {0}}$ having an extreme in it. The monotonicity of the function in $x_{0}$ ${\ displaystyle x_ {0}}$ will change.
If zero $x_{0}$ ${\ displaystyle x_ {0}}$ has an odd multiplicity of 3, 5, 7, etc., then, in view of $f'(x_{0})=0$ ${\ displaystyle f '(x_ {0}) = 0}$ , $f''(x_{0})=0$ ${\ displaystyle f '' (x_ {0}) = 0}$ and $f'''(x_{0})\neq 0$ ${\ displaystyle f '' '(x_ {0}) \ neq 0}$ , the function graph will have in $x_{0}$ ${\ displaystyle x_ {0}}$ inflection point (" saddle point "). The monotonicity of the function in $x_{0}$ ${\ displaystyle x_ {0}}$ will not change.

Literature

Schneider V.E. et al. Entire and fractional rational functions // A Short Course in Higher Mathematics. - M .: "Higher School", 1972. - S. 27-28. - 640 s.

Gelfand I.M., Glagoleva E.G., Shnol E.E. Polynomials // Functions and graphs (basic techniques). - M .: ICMMO , 2006 .-- 120 s. - ISBN 5-94057-131-X .

Lothar Papula. Ganzrationale Funktionen // Mathematik für Ingenieure und Naturwissenschaftler: [ him. ] . - Wiesbaden: Vieweg + Teubner, 2009 .-- T. 1. - S. 190-212. - ISBN 978-3-8348-0545-4 .

Stephan Bucher. Ganzrationale Funktionen höheren Grades // Anwendungsorientierte Mathematik für Techniker: [ him. ] . - München: Cal Hanser Verlag, 2016 .-- S. 106-114. - ISBN 978-3-446-44244-3 .

Links

Derivative of a power function // math24.ru