Given the function f (x) = x ^ 3 + 3x, for all t belonging to R f (T ^ 2-T) + F (3T ^ 2-k) > 0, the value range of real number k is

Given the function f (x) = x ^ 3 + 3x, for all t belonging to R f (T ^ 2-T) + F (3T ^ 2-k) > 0, the value range of real number k is

It is easy to know that f (x) is an odd function and an increasing function on (- ∞, + ∞)
If f (T ^ 2-T) + F (3T ^ 2-k) > 0 is constant, i.e
F (T ^ 2-T) > F (k-3t ^ 2) > 0 is constant, i.e
T ^ 2-T > k-3t ^ 2 is constant
4T ^ 2-T > k constant holds, i.e
And 4T ^ 2-T = (2t-1 / 4) ^ 2-1 / 16 ≥ - 1 / 16
So K

A mathematical problem, known 1 + A ^ 2 = 2B ^ 2, find the minimum absolute value of a-2b, and find the value of a, B at this time
A mathematical problem, known 1 + A ^ 2 = 2B ^ 2, find a-2b
The minimum value of absolute value of, and find the value of a, B at this time

Let t = a-2b. = = = > A = t + 2B. Substitute the equation into the question to get 1 + (T + 2b) & sup2; = 2B & sup2; = = = > 2B & sup2; + 4tb + 1 + T & sup2; = 0. ⊿ = 16t & sup2; - 8 (1 + T & sup2;) ≥ 0. = = = > T & sup2; ≥ 1. = = = > | t | ≥ 1. That is to say, | a-2b | ≥ 1. | a-2b | min = 1

If a and B are greater than zero and a + B + 1 = AB, what is the minimum value of 3A + 2B

A = (B + 1) / (B-1) can be obtained from a + B + 1 = ab,
Then b > 1 was obtained from a and b > 0
So 3A + 2B = (3b + 3) / (B-1) + 2B = [3 (B + 1) + 6] / (B-1) + 2B = 6 / (B + 1) + 2 (B-1) + 5 ≥ 2 radical 12 + 5 = 4 radical 3 + 5
If and only if 6 / (B-1) = 2 (B-1), that is, B = 1 + radical 3 "=" holds,
So the minimum of 3A + 2b is 4 radical 3 + 5

Given that ab = 3A + 2B, AB is greater than 0, then the minimum value of AB is?

Wrong..

Factorization of a ^ 3 + B ^ 3 + 3ab-1

Original formula = (a + b) ^ 3-3ab (a + b) + 3ab-1
=(a+b)^3-1-3ab(a+b-1)
=(a+b-1)[(a+b)^2+(a+b)+1]-3ab(a+b-1)
=(a+b-1)(a^2+b^2-ab+a+b+1)

Factorization of x ^ - 1 and 3AB ^ + A ^ B

Hello
x^-1=（x+1）（x-1）
3ab^+a^b=ab(3b+a)
If you don't understand, you can ask. If it helps, remember to adopt it
Thank you and wish you progress in your study!

Factorization of a Λ 2 + 3b-3ab-a

=a^-3ab-(a-3b)
=a(a-3b)-(a-3b)
=(a-3b)(a-1)
o(︶︿︶)o

Who can help me explain how factorization can change symbols?
For example, the following question
-ab(a-b)^2+a(b-a)^2-ac(a-b)^2
The answer I wrote in my textbook was (a-b) (AB + a-ac)
But the answer I'm getting older now is
(a-b)^2(ab+a+ac)

I don't know if you have the wrong number or something. The answer in the textbook is wrong
-ab(a-b)^2+a(b-a)^2-ac(a-b)^2==(a-b)^2*[-ab+a-ac)
Just mention the same term (a-b) ^ 2

Ask God to explain the symbol problem of factoring cross multiplication factor!
For example, 2x & # 178; - 5xy-42y & # 178; according to the absolute value, the sign of the larger factor is the same as that of the coefficient of the first term. Why is the answer not (2x-7y) (x + 6y)?

Note that the factor has a 2x & # 178; instead of X & # 178;;
② If the answer is (2x-7y) (x + 6y) = 2x & # 178; + 12xy-7xy-42y & # 178; = 2x & # 178; + 5xy-42y & # 178;, and the title is 2x & # 178; - 5xy-42y & # 178

Factoring 5 questions, the more the better, just answer a few
1) ( 2x^n-1 ) - ( 4x^2n-2 )
2) x(x-y)(3x+y) - 2x(x+y)(x-y)
3) 3 (a + b ) ^2 - 27c^2
4) (5m^2 +3n^2)^2 - (3m^2 + 5n^2)^2
5) x^4 - 18x^2 +81
I want to make sure that the more I answer, the better,

1) ( 2x^n-1 ) - ( 4x^2n-2 )=2X^n-1(1-X^n-1)2) x(x-y)(3x+y) - 2x(x+y)(x-y)=X(X-Y)[3X+y-2x-2y]=X(X-Y)(x-y)3) 3 (a + b ) ^2 - 27c^2=3[(a+b)^2-9c^2]=3(a+b+3c)(a+b-3c)4) (5m^2 +3n^2)^2 - (3m^2 + 5n^2)^2=[5...