It is proved that the function f (x) = x + X is a subtractive function on (0,1]

It is proved that the function f (x) = x + X is a subtractive function on (0,1]

x1,x2∈(0,1] x1>x2
f(x1)-f(x2)
=(x1+1/x1)-(x2+1/x2)
=(x1-x2)+(x2-x1)/(x1x2)
=(x1-x2)(1-1/(x1x2))
[x1>x2 x1-x2>0
Because x1x21
1-1/(x1x2)

prove; The function f (x) = 1 minus one tenth of X is an increasing function on (negative infinity, 0)

f(x)=1-1/x
Y = 1 / X is a subtractive function on (- infinity, 0)
Y = - 1 / X is an increasing function on (- infinity, 0)
F (x) = 1-1 / X is an increasing function on (- infinity, 0)
f(x)=1/(1-x)
x0
X increases - x decreases 1-x decreases 1 / (1-x) increases
So f (x) is an increasing function on (- infinity, 0)

It is proved that the function f (x) = x + X is a subtractive function on (- 1,0)

f(x)=x+1/x
f'(x)=1-1/x^2=(x^2-1)/x^2
When x ∈ (- 1,0), then x ^ 2

[high school mathematics] the function f (x) = 2Sin (x / 2 + π / 3) is known. If any x belongs to R, it has The function f (x) = 2Sin (x / 2 + π / 3) is known. If f (x1) < f (x2) < f (x3) for any x belonging to R, the minimum value of | x1-x2 | is _?

There is a problem with the input. It should be f (x1) ≤ f (x) ≤ f (x2) for any x belonging to R
Then f (x1) and f (x2) are the minimum and maximum values of F (x), respectively
Then the minimum value of | x1-x2 | is the half cycle T / 2 of the function
T=2π/(1/2)=4π,T/2=2π
That is, the minimum value of | x1-x2 | is 2 π
[Middle School Students' Mathematics and chemistry] the team will answer the questions for you

Known function f (x) = 1 / 2x ^ 2-ainx (a ∈ R) (1) If the tangent equation of function f (x) at x = 2 is y = x + B, find the values of a and B (2) If the function f (x) is an increasing function at (1, + infinity), find the value range of A (3) The number of solutions of equation f (x) = 0 is discussed

(1) The derivative of F (x) is f (x) '= x - A / x, and the slope of the tangent equation is f (2)' = 1. A = 2 can be obtained, then f (x) = 1 / 2x ^ 2-2inx, f (2) = 2 - 2in2. Substituting (2,2 - 2in2) into the tangent equation, B = - 2in2. (2) f (x) '= x - A / x, and f (x) is an increasing function at (1, + ∞), indicating that f (x)' ≥ 0 pairs

Known function f (x)= 3sin（2x-π 6）+2sin2（x-π 12） (1) Find the minimum positive period of function f (x); (2) Find the set of X when the function f (x) gets the maximum value

（1）f（x）=3sin（2x-π6）+1-cos（2x-π6）=2[32sin（2x-π6）-12cos（2x-π6）]+1=2sin（2x-π3）+1，∵ ω= 2，∴T=π； (2) Let 2x - π 3 = 2K π + π 2, K ∈ Z, the solution is: x = k π + 5 π 12, K ∈ Z, then the set of X when the function f (x) obtains the maximum value