Let f (x) = ln (x − 1) + 2aX (a ∈ R) (1) find the monotone interval of F (x); (2) if ln (x − 1) x − 2 > ax is constant when x ＞ 1 and X ≠ 2, then find the value range of real number a

(1) The definition domain of function f (x) is (1, + ∞), f ′ (x) = 1x − 1 − 2ax2 = x2 − 2aX + 2ax2 (x − 1). Let g (x) = x2-2ax + 2a, △ = 4a2-8a = 4A (A-2). ① when △ ≤ 0, i.e. 0 ≤ a ≤ 2, G (x) ≥ 0, ｜ f ′ (x) ≥ 0, f (x) monotonically increases on (1, + ∞). ② when a < 0, the axis of symmetry of G (x) is x = a, when x > 1, the function is quadratic (3) when a > 2, let x1, X2 (x1 < x2) be the two roots of the equation x2-2ax + 2A = 0, then X1 = a − A2 − 2A > 1, X2 = a + A2 − 2A. When 1 < x < X1 or x > X2, f '(x) > 0, f (x) is an increasing function on (1, x1), (X2, + ∞). When X1 < x < X2, f' (x) > 0, f (x) is an increasing function on (1, x1), (X2, + ∞) In conclusion, when a ≤ 2, f (x) increases monotonically on (1, + ∞); & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; When a ＞ 2, the monotone increasing interval of F (x) is (1, x1), (X2, + ∞), and the monotone decreasing interval is (x1, x2). (2) ln (x − 1) x − 2 ＞ ax can be reduced to 1 x − 2 [LN (x − 1) + 2 ax − a] > 0, that is, 1 x − 2 [f (x) − a] > 0, (*) let H (x) = f (x) - A, from (1): when a ≤ 2, f (x) is an increasing function on (1, + ∞), so h (x) is an increasing function on (1, + ∞) Because when 1 ＜ x ＜ 2, H (x) ＜ H (2) = 0, ■ (*) holds; when x ＞ 2, H (x) ＞ H (2) = 0, ■ (*) holds; so when a ≤ 2, (*) holds; when a ＞ 2, because f (x) is a decreasing function on (x1, 2), H (x) is a decreasing function on (x1, 2), so when x1 ＜ x ＜ 2, H (x) ＞ H (2) = 0, (*) does not hold In conclusion, the value range of a is (- ∞, 2]

Is the image of F (x) = ln (X-2) a decreasing function or an increasing function
I am puzzled because there is a problem
Of the following functions, the one that is a decreasing function in its domain is
A f（x）=-x^2+x+1
B.f(x)=1/x
C.f(x)=1/3^IxI
D.f(x)=ln(x-2)
The answer given by the teacher is D, very confused

Sophomore of it! (I found that you are one day younger than one of my classmates, he December 11) don't know, ask me again!
Increasing function
Let t = x - 2, we know that t is an increasing function
And ln (T) is also an increasing function, so increasing is increasing, so f (x) is an increasing function
Wrong topic, wrong teacher
A wrong, because there is a decrease and an increase
B wrong, because it can't be said to be on the domain
C wrong, because there is an increase and a decrease
D wrong, because it's an increase

What is the 33rd digit after the decimal point of Pi?

3. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 4811174502 8410270193 8521105559 6446229489 5493038196
The above is pi, you can count to 33

(x-1) ^ 2cosnpix integral PI = Pi

∫(x-1)^2cosnpixdx
=1/(npi)∫(x-1)^2d(sinnpix)
=1/(npi)(x-1)^2sinnpix+2/(npi)^2∫(x-1)d(cosnpix)
=1/(npi)(x-1)^2sinnpix+2/(npi)^2(x-1)cosnpix-2/(npi)^2∫cosnpixdx
=1/(npi)(x-1)^2sinnpix+2/(npi)^2(x-1)cosnpix-2/(npi)^3sinnpix+C
=(x-1)^2sinnpix/(npi)+2(x-1)cosnpix/(npi)^2-2sinnpix/(npi)^3+C

How to calculate pi?

In the middle of the third century, Liu Hui, a mathematician in the Wei and Jin Dynasties, initiated the circle cutting technique, which established a rigorous theory and a perfect algorithm for calculating the PI. The so-called circle cutting technique is a method to calculate the circumference of a circle by continuously multiplying the number of sides of the inscribed regular polygon
Liu Hui invented "cutting circle" to seek "Pi". What does PI really mean? It actually refers to "the ratio of the circumference of a circle to the diameter of the circle". Fortunately, it is a constant! With it, we can calculate all kinds of circles and spheres. Without it, we will have nothing to do with circles and spheres, The accuracy of PI value is also directly related to the accuracy and accuracy of our calculation. This is why human beings require PI and it is accurate
According to "circumference / diameter = circumference ratio", then circumference = diameter * circumference ratio = 2 * radius * circumference ratio (this is the reason why we are familiar with circumference = 2 π R). Therefore, there is no need to memorize the "circumference formula". As long as you have primary school knowledge and know the meaning of "circumference ratio", you can deduce and calculate it by yourself, But its "meaning and function" is often ignored, which is the meaning of cyclotomy. Because "circumference ratio = circumference / diameter", the "diameter" is straight and easy to measure; the "circumference" is difficult to calculate accurately. Through Liu Hui's "cyclotomy", this difficult problem can be solved. As long as the circumference is calculated carefully and patiently, the "diameter" is easy to measure, As we all know, Zu Chongzhi finally finished this work in China

How to calculate the Pi? Write the calculation method!

The ancients used the method of cutting circle to calculate the circumference of a circle. Archimedes used regular 96 polygon to get the precision of 3 decimal places; Liu Hui used regular 3072 polygon to get the precision of 5 decimal places; Rudolph used regular 262 polygon to get the precision of 35 decimal places

What is the sum of the first digit after the decimal point of PI added to the 2010 digit?

1+4+1+5+9+2+6+5+3+5+8+9+7+9+3+2+3+8+4+6+2+6+4+3+3+8+3+2+7+9+5+0+2+8+8+4+1+9+7+1+6+9+3+9+9+3+7+5+1+0+5+8+2+0+9+7+4+9+4+4+5+9+2+3+0+7+8+1+6+4+0+6+2+8+6+2+0+8+9+9+8+6+2+8+0+3+4+8+2+5+3+4+2+1+1+7+0+6+7+9+...

The square of 1 / 2 π r = 3 π, then the square of R =, according to

R squared = 6
If you multiply two sides of the equation at the same time, the equation still holds

Let f (x) = ln (x − 1) + 2aX (a ∈ R) (1) find the monotone interval of F (x); (2) if ln (x − 1) x − 2 > ax is constant when x ＞ 1 and X ≠ 2, then find the value range of real number a