Let the domain of definition of function f (x) be D, and there exists x0 ∈ D, so that f (x0) = x0 holds. The point with (x0, x0) as the coordinate is called the fixed point on the image of function f (x) 1. If the function f (x) = 3x + A / x + B (a, B ∈ R) has two symmetric fixed points about the origin, find the conditions that real numbers a and B satisfy; 2. Is the proposition "if there are finite fixed points on the image of odd function f (x) defined on R, then there are odd fixed points?" correct? Give proof or give counterexample. What is the conclusion about even function?

Let the domain of definition of function f (x) be D, and there exists x0 ∈ D, so that f (x0) = x0 holds. The point with (x0, x0) as the coordinate is called the fixed point on the image of function f (x) 1. If the function f (x) = 3x + A / x + B (a, B ∈ R) has two symmetric fixed points about the origin, find the conditions that real numbers a and B satisfy; 2. Is the proposition "if there are finite fixed points on the image of odd function f (x) defined on R, then there are odd fixed points?" correct? Give proof or give counterexample. What is the conclusion about even function?

1. In essence, when f (x) - x = 0, there are two roots x1, X2, and X1 + x2 = 0
F (x) - x = 0 can be reduced to 2x ^ 2 + BX + a = 0 (x is not equal to zero)

Do the image of the function f (x) = - x ^ 2 + X, and find the monotone interval

Draw your own picture, about Y-axis symmetry
Monotone interval:
Monotonically increasing on [- 1 / 2, -∞) and [0,1 / 2) respectively,
Monotonically increasing on (0,1 / 2) and (1 / 2, + ∞) intervals respectively,
Note: it is not monotone increasing on [- 1 / 2, -∞) ∪ [0,1 / 2) interval, because it is neither continuous interval nor monotone

Let y = (x ^ 2 + 3) / (x-1) answer the following questions: 1) divide the monotone interval of the function and find the extremum; 2) divide the concave of the graph of the function
section
3) Seeking asymptote of graph

Domain x is not equal to 1
Derivative, so that the reciprocal is 0, x = - 1,3
Simple increase (- infinity, - 1] u [3, + infinity)
Single minus (- 1,1) U (1,3)
The positive interval is concave and the negative interval is convex

The graph of function y = | X-2 | (x + 1) is given, and the monotone interval of function is found according to the graph of function

Y = | X-2 | (x + 1) = x2 − x − 2, X ≥ 2 − x2 + X + 2, x < 2, so the image of the function is a combination of some parts of two quadratic functions. According to the image method of quadratic function, we can make the image of the function. Note that the symmetry axis of the quadratic function where the two images are located is x = 12, as shown in the following figure: from the image, we can get the monotone interval of the function: monotone recurrence The increasing interval is: (- ∞, 12), (2, + ∞), and the decreasing interval is (12, 2)

Given that the lengths of two sides of a right triangle are 3 and 4 respectively, find the length of the third side. (with Pythagorean theorem)

√3²+4²=5
Or √ 4 & # 178; - 3 & # 178; = √ 7

The length of the two right sides of a right triangle is 3cm and 4cm respectively, and the length of the hypotenuse is 5cm. The height of the hypotenuse of the right triangle is ()
A. 3.6cmB. 2.4cmC. 1.2cm

A: the height of the hypotenuse of this right triangle is 2.4cm

How does the hypotenuse change when the two right sides of a right triangle are doubled? When the two right sides of a right triangle are doubled, the hypotenuse will change

It has also tripled
Because right triangle Pythagorean theorem
A^2+B^2=C^2
(2A)^2+(2B)^2=4(A^2+B^2)=4C^2

How does the hypotenuse change when the two right angles of a right triangle are enlarged twice?

Zoom in 2 times
Simple (special triangle) example: the original triangle edge is 3-4-5, and the enlarged triangle edge is 6-8-10

Are two right triangles corresponding to a right angle and a hypotenuse similar? Please explain why

Yes, let a RT triangle have two right angled sides a and B, and the hypotenuse C. in this way, we can know the length proportion of the other side through a ^ 2 + B ^ 2 = C ^ 2, so the three sides are proportional, so they are similar

The relationship between the hypotenuse and the height of the hypotenuse of a right triangle and two right angles and its proof
Correct answer + 50 points, give a triangle ABC, angle c is 90 degrees, make CD perpendicular to AB, verify: AC + BC < CD + BC

It should be AC + BC < CD + AB (AC + BC) & sup2; - (CD + AB) & sup2; = AC & sup2; + BC & sup2; + 2Ac * bc-cd & sup2; - AB & sup2; - 2CD * AB Pythagorean theorem AC & sup2; + BC & sup2; = AB & sup2; and triangle area = AC * BC / 2 = AB * CD / 2, so 2Ac * bc-2ab * CD = 0, so (AC + BC) & sup2; - (CD + AB