Double hook function

Nike function

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Synonym double hook function generally refers to hook function.

Hook function is a general hyperbolic function similar to inverse proportional function, and its form is f (x) = ax+b/x (a >; 0，b & gt0).

Chinese name

Nike function

Another name

Hook function, fishhook function, Nike function, double hook function, inspection function, Shuang Yan function.

express

f(x)= ax+b/x(a & gt； 0，b>0)

Applied discipline

mathematics

area of application

algebraic function

area of application

Analytic geometry

catalogue

1 definition

definition

name

2 attributes

draw

Most valuable

Parity and monotonicity

Asymptote

3 inequality between the minimum value and the average value of hook function

4 derivative solution

5 Other solutions

Six key points

Seven examples

definition

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definition

The so-called hook function (hyperbolic function) shape image

(a & gt0).

name

Named for its image, it is also called "double hook function", "hook function", "stop function" and "Shuang Yan function". Also known as "Nike function" or "Nike curve".

nature

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draw

Tick function is a common and special function in mathematics. See the picture. You'd better draw an asymptote when drawing.

Most valuable

When x> is 0,

There is a minimum value (here, a>0, b>0 for the convenience of research). When is this?

F(x) takes the minimum value.

Parity and monotonicity

odevity

The double hook function is an odd number function.

monotonicity

Make k=

, and then:

Interval increase: {x|x≤-k} and {x | x≥k }；; Negative interval: {x |-k ≤ x

Trend: the left side of Y axis increases first and then decreases, and the right side of Y axis decreases first and then increases, which are two hooks.

Asymptote

The image of hook function is two curves with Y axis and y=ax as asymptotes respectively, any one on the image.

Nike function

The product of the distance from a point to two asymptotes is exactly the product of the sine value of the included angle of the asymptotes (0- 180) and |b|.

Note: the image of tick function is a hyperbola. In fact, the image is axisymmetric and can be obtained by the standard equation of hyperbola through the rotation angle.

Inequalities of Minimum and Average Value of hook Function

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The study on the properties of hook function is inseparable from the mean inequality. Speaking of mean inequality, it is actually based on quadratic function. We all know

Expand, acquire

, namely

Add 2ab on both sides at the same time, and it's done.

When the two sides are squared, the formula of the mean value theorem is obtained:

will

middle

Think of it as a,

When b is substituted into the above formula, you must

There is a rule here: if and only if ax=b/x, find the minimum value and solve x=sqrt(b/a), and the corresponding f(x)=2sqrt(ab). Let's look at the mean inequality again. It can also be written as: (a+b)/2≥sqrt(ab). As we all know, the former formula is the average formula. What about the latter formula? It is also the formula of average, but the difference is that the former is called arithmetic average and the latter is called geometric average. To sum up, the arithmetic average will never be less than the geometric average.

Derivative solution

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In fact, derivatives can also be used to study the properties of hook functions. But first you have to know the conversion of negative exponential power, which is also very simple, but you have to master it skillfully. Give a few examples: 1/x = x- 1, 4/x 2 = 4x-2. When x is the denominator, it can be converted into a negative exponential power. Then f(x) = ax+b/x = ax+bx- 1, and the derivative function is a+(-b) x-2, so that f'(x)=0, and b = ax 2 is calculated, and the result is still x = sqrt (b bSee which one you like to use when doing problems, derivative or median. However, pay attention to the mean value theorem discussed at last. Sometimes ax≠b/x can't be used. Maximum and minimum values on (you can use your research conclusions).

When x>0, f(x)=ax+b/x has a minimum value; When x

f(x)=x+ 1/x

First of all, you should know that his domain is x, which is not equal to 0.

When x>0,

According to the average inequality:

f(x)= x+ 1/x & gt； =2 root number (x* 1/x)=2

When x= 1/x is equal.

X= 1, the minimum value is 2, and there is no maximum value.

When x

f(x)=-(-x- 1/x)& lt； =-2

When -x =- 1/x is equal.

X=- 1, maximum -2, no minimum.