I'm not good at high math, 1. How does the auxiliary function used in the proof of Lagrange mean value theorem come from? The auxiliary function is: F (x) = f (x) - {f (a) + [f (b) - f (a)] / (B-A) * (x-a)} 2. Derivation process of elementary function (proof of derivative function of elementary function) thank you~ 3. The derivatives of inverse functions are reciprocal to each other, so why don't the logarithm of X with a as the base and the x power of a be reciprocal to each other? If you know, please reply more~

I'm not good at high math, 1. How does the auxiliary function used in the proof of Lagrange mean value theorem come from? The auxiliary function is: F (x) = f (x) - {f (a) + [f (b) - f (a)] / (B-A) * (x-a)} 2. Derivation process of elementary function (proof of derivative function of elementary function) thank you~ 3. The derivatives of inverse functions are reciprocal to each other, so why don't the logarithm of X with a as the base and the x power of a be reciprocal to each other? If you know, please reply more~

1. This auxiliary function is introduced to achieve the condition f (a) = f (b) of Rolle's mean value theorem (if you substitute a and B, then all f (x) are equal to 0), so according to Rolle's mean value theorem, there must be a point ξ on (a, b) satisfying f '(ξ) = 0. If f' (x) = f '(x) - [f (b) - f (a)] / (B-A) is obtained by deriving f (x), that is, f' (ξ) / (B-A)

Prove how the function constructed by Lagrange's mean value theorem is thought out

Because we have learned Rolle's theorem before, we need to apply Rolle's theorem when proving Lagrange's theorem, so we need a constructor to satisfy the conditions of Rolle's theorem. The constructed function is not unique, as long as it can satisfy the conditions of Rolle's theorem

How to understand the auxiliary function of proving Lagrange's mean value theorem
The auxiliary function is: F (x) = f (x) - {f (a) + [f (b) - f (a)] / (B-A) * (x-a)} the process of obtaining this function

In fact, this auxiliary function f (x) = f (x) - H (x)
H (x) = {f (a) + [f (b) - f (a)] / (B-A) * (x-a)} & nbsp; is a straight line passing (a, f (a)) and (B, f (b))
In other words, it is a curve f (x) & nbsp; (a means (a, f (a)), B means (B, f (b)))
Subtract the line H (x) connecting two points ab
I have a picture attached
In the graph, the curve is f (x), the straight line is h (x), and the subtraction of two functions is f (x)
The intuitive image of Lagrange mean value theorem is the tangent of a point in the image
(see the dotted line in the figure) make the tangent parallel to AB, which can be seen intuitively from the figure
The auxiliary function is used in the proof process to change the Lagrange mean value theorem into Rolle mean value theorem
You can see that after subtracting H (x), f (x) becomes f (x), (let's write H (x) as KX + L)
Then the derivative of F (x) = the derivative of F (x) - K
K is the slope of the line between two points ab. to prove that there is a point s such that f '(s) = k, that is to prove that there is a point s such that f (s) = 0
Because f is 0 at x = A and x = B, it becomes the case of Rolle's mean value theorem
What questions can I ask you

In the space rectangular coordinate system, find a point C on the Z axis so that the distance from point C to point a (1, 0, 2) is equal to point B (1, 1, 1), then the coordinate of point C is______ ．

Let C (0, 0, z) have the same distance from point C to point a (1, 0, 2) and point B (1, 1, 1), then we get 12 + 02 + (Z-2) 2 = 12 + 12 + (Z-1) 2, and the solution is Z = 1, so C (0, 0, 1) so the answer is: (0, 0, 1)

If the right branch of the straight line L: y = KX + 1 and hyperbola C: 2x2-y2 = 1 intersect at two different points a and B, then the value range of the real number k is ()
A. − 2 ＜ K ＜ 2B. - 2 ＜ K ＜ 2C. K2 ＜ 4 and K2 ≠ 2D. - 2 ＜ K ＜ 0 and K ≠ − 2

After substituting the equation y = KX + 1 of the straight line l into the equation 2x2-y2 = 1 of the hyperbola C, we get (k2-2) x2 + 2kx + 2 = 0. According to the meaning of the problem, the right branch of the hyperbola C and the straight line L intersect at two different points. Let the coordinates of the two intersections be (x1, Y1), (X2, Y2), (x1, Y1), (X2, Y2) at the right branch of the hyperbola C,  x1 ＞ 0, X2 ＞ 0, ｜ X1 + x2 = − 2K & nbsp; K2 − 2 ＞ 0, x1x2 = 2k2 − 2 ＞ 0, so K2 − 2 ≠ 0 △ = (2k) 2 − 8 (K2 − 2) ＞ 0 − 2kk2 − 2 ＞ 02k2 − 2 ＞ 0. The value range of K is - 2 ＜ K ＜ 2, so a

5 () 5 () 5 () 5 = 24, fill in the operation symbol in the bracket, and the bracket

5 × 5 - 5 ÷ 5 = 24

Find the number of real roots of log2 (x + 4) = 3 ^ X
Log 2 (x + 4) = the number of real roots of 3 ^ X

This is a high school logarithmic and exponential function problem, this kind of problem is very classic in high school, because according to the high school knowledge can not solve the base and index, there are unknowns, so the most commonly used method is to draw the function of the equation, and then observe the distribution and number of roots (the intersection is the root)

1. It is known that the sum of solutions X and y of the system X -- y = 2A, X -- 3Y = 1 -- 5A is a positive number, and the value range of a is obtained
2. Given X & sup3; + X & sup2; + X + 1 = 0, find the power of 1 + X + X & sup2; + X & sup3; +. + X
3. In the right triangle ABC, the three sides are a, B, C, where C is the hypotenuse. If B divided by C + A and a divided by C + B equals 17 / 20, find a: B: C (no semicolon)
4. Explain why (x + 1) (x + 2) (x + 3) (x + 4) + 1 is a complete square expression
5. When a leaky wine jar is full of wine, three people can drink it for six days, five people can drink it for four days. How many days can one drink it for?
Brothers and sisters, you can help me with a few questions,

1. In general, such a system of equations must be solved first
The results show that x = (11a-1) / 2, y = (7a-1) / 2
x+y=9a-1＞0
∴a＞1/9
2. From the beginning, every four were divided into groups,
The original formula = (1 + X & sup1; + X & sup2; + X & sup3;) + x ^ 4 (1 + X & sup1; + X & sup2; + X & sup3;) + x ^ 8 (1 + X & sup1; + X & sup2; + X & sup3;) +... + x ^ 2000 (1 + X & sup1; + X & sup2; + X & sup3;)
=0
(every four terms of the formula add up to zero. There are 2004 terms in total, and there are 501 groups.)
3. 8:15:17 or 15:8:17
This is an experiment, and the process is not easy
4. Certification:
Original formula = [(x + 1) (x + 4)] [(x + 2) (x + 3)] + 1
=(x²+5x+4)(x²+5x+6)+1
=[(x²+5x+5)-1][(x²+5x+5)+1]+1
=(x²+5x+5)²-1²+1
=(x²+5x+5)²
Therefore, the original formula is always a complete square number
5. Let one drink X and leave out y in one day
6(3x+y)=4(5x+y)
x=y
All of them are expressed by X. the original wine in the jar is 6 × 4x = 24x
The daily consumption is 2x
It takes 24x △ 2x = 12 days to drink as much as possible

What's the English of mathematical number set symbols, such as what's the English of N
It means those R n Q those

Man
N is the set of natural numbers, abbreviated as n
R is a set of real numbers. As the name suggests, it is any number
Q is a set of rational numbers
Set of n *. Positive integers
Z is a set of integers

What's the difference between pinyin "in" and "ing"?

In front of the nasal sound, the sound with "Yin", ing after the nasal sound, with "Eagle", this thing is really not easy to use words to describe, ha ha, do not know you understand it?
Remember to adopt it