Knowing 1 n 2 1 m 2 a 2 b 2 a 2 b 2 , find the maximum value of 1 2mn 10

Hello! This question is a question to investigate the basic properties of inequalities, and the answer is as follows, 1 n 2+1 m 2=(1 n) 2+(1 m) 2>=2*(1 n)*(1 m), so 1 mn<=((1 n) 2+(1 m) 2) 2=(a 2+b 2) (2*a 2*b 2), multiply 1 2mn by 1 2 on the left and right, and get 1 2mn<=(a 2+b 2) (4*a 2*b 2), so the maximum value of 1 2mn is (a 2+b 2) (4*a 2*b 2), I wish you good progress!

Because m 2 n 2 11, then return to the hall respectfully.

m＋n）＾2＝m＾2＋n＾2＋2mn

11 Drain 2mn 11 m 2 n 2

11 11 22 (take the equal sign when m n).

Thus m volts rent n丨max 22, so, (m n) max 22.

mn-n²=1

Get 2mn-2n hand striker friend = 2

m -n Bi Huai = 4

m -2mn + n = m -n kibi - (2mn-2n ) = 4-2 = 2

Original formula =-2m 2+2mn-2n 2-n 2+m 2-2mn-m 2-3n 2=-(m 2+3n 2)m 2+3mn=5, (1).

3n^2-3mn=61×3 （2）

Infiltrate 1) + (2) early bond to obtain plexus fluid.

m^2+3n^2=5+183=188

So the original formula -(m 2+3n 2).

Knowing =n+2, n =m+2 m≠n , then:

m²-n²=n-m

i.e. (m-n)(m+n)=-(m-n).

Since m≠n, we get: m+n=-1

Then: m +n = n + 2 + m + 2 = m + n + 4 = 3 and 2mn = (m + n) -m + n ) ) = 1-3 = -2 then we get: mn = -1

So: m -2mn+n

m+n)(m²-mn+n²)-2mn

(3-mn)-2mn-3-mn

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m²=n+2

n²=m+2

Subtract the two formulas.

m²-n²=n-m

m-n)(m+n)=-(m-n)

Since m-n is not equal to 0, both sides are divided by (m-n)m+n=-1 at the same time

m³-2mn+n³

m³-mn+n³-mn

m(m²-n)+n(n²-m)

2m+2n2(m+n)

Knowing =n+2, n =m+2 m≠n , then:

m²-n²=n-m

i.e. (m-n)(m+n)=-(m-n).

Since m≠n, we get: m+n=-1

Then: m +n = n + 2 + m + 2 = m + n + 4 = 3 and 2mn = (m + n) -m + n ) ) = 1-3 = -2 then we get: mn = -1

So: m -2mn+n

m+n)(m²-mn+n²)-2mn

3-mn)-2mn-3-mn

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m²=n+2

n²=m+2

Subtract the two formulas.

m²-n²=n-m

m-n)(m+n)=-(m-n)

Since m-n is not equal to 0, both sides are divided by (m-n)m+n=-1 at the same time

m³-2mn+n³

m³-mn+n³-mn

m(m²-n)+n(n²-m)

2m+2n2(m+n)=-2

m = n+2, n = m+2, m≠n , subtract to get.

m²-n²=n-m

m-n][m＋n]=n-m

m＋n=-1

Multiply, get. m²n²=mn＋2【m＋n】＋4

mn】²-mn-2=0

mn＋1】【mn-2】=0

mn=-1 or mn=2 [rounded].

n = m+2, m≠n multiply, get.

m²n²=mn＋2【m＋n】＋4

mn】²-mn-2=0

mn＋1】【mn-2】=0

mn=-1 or mn=2 [rounded].

m³-2mn+n³

m³+n³-2mn

m＋n]【m²-mn＋n²】-2mn

m²＋mn-n²-2mn

【m＋n】²＋mn-1＋mn

(1) m = n+2(a), n = m+2(b)(a)-(b) get (m+n)(m-n)=n-m, because m≠n

So m+n 1

2) Be observant.

m = n+2 multiplied by m on both sides at the same time

i.e. m nm 2m (a).

n = m+2 multiplied by n at the same time

i.e. n mn 2n (b).

a) (b).

m³-2mn+n³＝2（m+n）＝－2

Title: Manuscript has an M doesn't equal N, right? Balance rolling

m²=n+2

n²=m+2

Subtract the two formulas.

m²-n²=n-m

m-n)(m+n)=-m-n)

Since m-n is not equal to 0, both sides are divided by (m-n)m+n=-1 at the same time

m³-2mn+n³

m -mn+n commanding respect-yu-mn

m(m²-n)+n(n²-m)

2m+2n2(m+n)