Seeking solutions, the whole process, high school mathematics

Teenagers, this topic is typical.

Function monotonicity. How many ways can you think of getting the monotonicity of a function?? The first definition of monotonicity, which can be used to prove the monotonicity of this function according to the definition of monotonicity.

Second. The derivative of the function, the derivative greater than or equal to 0 is the increasing function, and less than or equal to 0 is the decreasing function. Third, common sense, according to the graph of a typical function or the graph of a function obtained by the displacement transformation of a typical function, there is usually a translation up and down, and a translation to the left and right.

to determine the monotonicity of the function.

When you encounter a question, don't come up and start asking for the answer. This is not beneficial for later learning. You can only analyze the topic by yourself and review all the knowledge points used.

Here are the answers for you. There are 2 ways to do this, and the definition is proof. I'm not going to prove it to you here.

You can prove it yourself by consulting the textbook. Method 2 is proved by the derivative of the function. This method is generally simpler.

Through analysis, the monotonicity of a function can be obtained by the derivative of the function.

Answer: f(x)=-1 x then f'(x)=(-1/x)'=-(x^(-1))'=1 x 2 x is (0, positive infinity), so in this interval 1 x 2 is Evergrande to 0. i.e. f'(x) is a monotonically increasing function from 0 to positive infinity.

The title has been proven.

The graph of the function, you can see for yourself. Give you ideas. Start by drawing a graph of f(x)=1 x.

Let's talk about the x-axis symmetry, and the resulting graph is the graph of ff(x)=-1 x. You can also see from the graph that this function is monotonically increasing.

Note that x ranges from 0 to positive infinity.

Looking forward to helping you. Think about it yourself next time and ask. If you don't understand, you can continue to ask. Good luck. Think about this topic repeatedly.

Because x is increasing at (0, plus infinity). So, f(x)=1 x is a monotonically decreasing function at (0, positive infinity).

Study hard, boy, this is the foundation!

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1. Let the edge length of the cube ABCD-A1B1C1D1 be 1, and find the height of the triangular pyramid B-Ab1C.

The bottom is a lower triangle with a base of 2, a triangular pyramid with an edge of 1, and a height of 3 3.

2.Find the surface area and volume of the regular triangular pyramid P-abc with a base edge length of 2 root number 3 and a side edge length of root number 5.

The height is 1, the base area: 3 3, the side area: 6, the surface area: 3 3 + 3 6. Volume: 3

3.There is an inscribed cylinder inside a hemisphere, one base of the cylinder is on the plane of the hemisphere, and the other bottom is round on the sphere, the radius of the ball is r, find the maximum value of the side area of the cylinder.

Let the height of the cylinder be r=rcos, the radius be h=rsin, and the side area s=2 rh=2 r 2*cos *sin = r 2*sin2 = r 2, (2 = 2, take the maximum value).

4.There is a cube in the hemisphere, 4 vertices are on the plane of the hemisphere, and the other 4 vertices are on the sphere, find the ratio of the full area of this hemisphere to the full area of the cube.

The figure is completed into a sphere on the plane of the hemisphere, and a rectangle composed of two squares, the diagonal of the rectangle is the diameter of the ball, obtained by the Pythagorean theorem: (2a) 2+( 2a) 2=(2r) 2, r = 6a 2, the ratio of the full area of the hemisphere to the full area of the cube = 3r 2 6a 2=3 4

Let me give you a brief idea.

1.Establish a coordinate system with a cube to find the coordinates of the points on the bottom surface of the prism, and any two points can obtain a vector a1 and a2. Let the normal vector of the surface be n.

With a1xn=0 a2xn=0, we can get n. The equation of the surface can be obtained by the point French equation, which is reduced to a general formula, using the distance formula from the point to the surface: the surface ax+by+cz+d=0 and the point(x,y,z) point-to-surface distance =|ax+by+cz+d|(under the root number (a 2 + b 2 + c 2)) to get the result.

The first question is 2 points and 2 points

1、b.Translate y=f(x+1) 1 unit to the right to get y=f(x). Thus y=f(x) is monotonically decreasing and passing (1,0) points. So 1,4 is correct.

2、a。Definition of monotonicity.

3、d。Because the function is monotonous, and the values of the functions at both ends are positive and negative, it must be crossed once on the x-axis. So there is an x such that y=0 (provided that the function y=f(x) must be continuous and uninterrupted).

4、c。It is derived from the image and properties of a quadratic function. The axis of symmetry x = 3, which crosses the interval (2,4).

5、c。Because I didn't say x16, a. First, define the domain. x 2+2x-3>=0, we get x<=-3, or x>=1. x 2+2x-3 is monotonically reduced on (- 3), so it is still monotonically reduced after the root number is opened.

7、m>0。Since the function is subtracted over r, f(m-1) > f(2m-1), giving m-1<2m-1. So m>0

8、-3。From the meaning of the title, x=2 is the axis of symmetry, so m=8So f(x)=2x2-8x+3So f(1)=2-8+3=-3.

9. The monotonic increase interval is (- 1] and [0,1], and the monotonic decrease interval is (-1,0], [1,+].

The function image is as follows:

1.The trajectory equation for point A is: x 2 + y 2 = 12

ap|^2 = (x + n)^2 + y^2 = (x^2 + y^2) +2n + n^2 = 1 + 2n + n^2

aq|^2 = (x - n)^2 + y^2 = (x^2 + y^2) -2n + n^2 = 1 - 2n + n^2

pq|^2 = 4n^2

So, |ap|^2 + aq|^2 + pq|^2 = 2 + 6n^2。Visible is a fixed value.

The derivative, while the function e x is equal to mx at the tangent point.

Syndicate, get m=1

spacb=pa*bc

When the PC is perpendicular to KX+Y+4=0, the PC is the smallest, so that the PB is the smallest.