If x ^ 2 + 2x + 2A = 0 and x ^ 2 + 2 (2-A) x + 4 = 0 have and only one equation has two unequal real roots, then the value range of real number a is

Equation 1: x ^ 2 + 2x + 2A = 0 and equation 2: x ^ 2 + 2 (2-A) x + 4 = 0 have and only one equation has two unequal real roots. When one equation has a solution and only has different roots, the other equation has no solution or has a real root. Equation 1: △ = 4-8a, equation 2: △ = 4 (A-2) ^ 2-16 = - 4 [(A-2) ^ 2-4] has two

The equation (A-2) x ^ + (1-2a) x + a = 0 of X has real roots, so we can find the value range of A

(A-2) x ^ 2 + (1-2a) x + a = 0 has real roots,
(1)a=2,x=2/3
(2)a≠2,
Δ=4a+1>=0
a>=-1/4
A ≥ - 1 / 4

Let f (x) be an odd function defined on R, and f (x + 2) = f (- x) (x belongs to R). It is proved that f (x) is a periodic function

F (x + 4) = f (x + 2 + 2) = f (- X-2) = - f (x + 2) = - (- f (x)) = f (x), where the third equal sign is because f is an odd function, so 4 is the period of F

The function f (x) is an odd function defined on R, f (x + 2) = - f (x), which is proved to be a periodic function
How to prove it?

Prove: ∵ f (x + 2) = - f (x)
∴f(x+4)=-f(x+2)=-[-f(x)]=f(x)
F (x) is a periodic function

The symmetry axis of image with function y = 2 / (TaNx Cotx)
The answer to this question is (- π / 8,0),

Let the axis of symmetry be (T, 0), then
2/(tan(x-t)-cot(x-t))=2/(tan(2t-x)-cot(2t-x));
tan(x-t)-cot(x-t)=tan(2t-x)-cot(2t-x);
tan(2t-x)-tan(x-t)=cot(2t-x)-cot(x-t);
Right = [Tan (x-t) - Tan (2t-x)] / [Tan (x-t) · Tan (2t-x)];
Then Tan (2t-x) - Tan (x-t) = [Tan (x-t) - Tan (2t-x)] / [Tan (x-t) · Tan (2t-x)];
tan(x-t)·tan(2t-x)=-1;
tan(x-t)·tan(x-2t)=1;
[(tanx-tant)/(1+tanx·tant)][(tanx-tan2t)/(1+tanx·tan2t)]=1;
(tanx)^2-(tant+tan2t)tanx+tant·tan2t=1+tanx·(tant+tan2t)+(tanx)^2·tant·tan2t
(tanx)^2·(1-tant·tan2t)-2tanx·(tant+tan2t)=1-tant·tan2t;
[1-(tanx)^2]·(1-tant·tan2t)+2tanx·(tant+tan2t)=0;
[1-(tanx)^2]·{1-tant·2tant/[1-(tant)^2]}+2tanx·[tant+2tant/(1-(tant)^2]=0;

In the four functions of y = TaNx, y = sin (x + Wu / 2), y = sin2x, y = sin (2x Wu / 2), the number of increasing functions on the interval (0, Wu / 2) is

In the four functions, ① y = ∣ TaNx ∣; ② y = ∣ sin (x + π / 2) ∣; ③ y = ∣ sin2x ∣; ④ y = sin (2x - π / 2), the number of increasing functions on the interval (0, π / 2) is 1
Only y = ∣ TaNx ∣, which is not only a even function with π as a period, but also an increasing function on (0, π / 2)

Is y = x * TaNx a periodic function? What is the minimum positive period

Solving derivative problem

If the function y = f (x) and the image of X belonging to R are symmetric with respect to the line x = A and x = B (b > A), it is proved that f (x) is a periodic function and 2 (B-A) is one of its periods
How to get f (x) = f (2a-x) and f (x) = f (2b-x) from the symmetry of two lines and how to solve the problem correctly

I see. Thank you

Let the graph of function y = f (x), X ∈ (- ∞, + ∞) be symmetric with respect to x = a, x = B, (a < b), and prove that y = f (x) is a periodic function
What conclusion can we draw when y = f (x), X ∈ (- ∞, + ∞) graphs are symmetric with respect to x = a, x = B? Why?

For x = a symmetry, f (a + x) = f (A-X)
For x = B symmetry, f (B + x) = f (b-X)
f(x)=f[a+(x-a)]=f[a-(x-a)]=f(2a-x)=f[b+(2a-x-b)]=f[b-(2a-x-b)]=f[x+2(b-a)]
t=2(b-a)

If x ^ 2 + 2x + 2A = 0 and x ^ 2 + 2 (2-A) x + 4 = 0 have and only one equation has two unequal real roots, then the value range of real number a is