Knowing the positive real number a, b satisfies ab a b 3, find the minimum value of a b and the mini

If a, b are positive real numbers.

satisfying ab=a+b+3, find the range of ab.

Solution: a>0,b>0, ab=a+b+3>3

Let ab=u, then b=u a, substitute ab=a+b+3, and get:

U=A+U A+3=(A+3A+U) A, so A+(3-U)A+U=0

Since a is a real number, it is discriminative.

3-u) -4u=u -10u+9=(u-9)(u-1) 0 gives u 9 or u 1 (rounded off, because u>3 is known) when u=ab=9, a+b=6, and a=b=3

In other words, the value range of ab is [9,+

The value range of a+b is [6,+

The minimum value of a+b is equal to 6

The minimum ab value is equal to 3

It is to use a 2 + b 2 > = 2ab

The minimum value of a+b is 6, and the minimum value of ab is 9

a+b=ab+3≥(ab)½

ab)½ 3)((ab)½ 1 )≥0

ab)½ 3

ab≥9ab)½≤a+b)/2

ab≤(a+b)²/4

a+b)²-4(a+b)-12≥0

a+b-6)(a+b+2)≥0

a+b≥6

The idea of commutation is that t=a+b, and then put the inequality, a+b >= 2 root number ab=2 root number (a+b+3), the square of both sides gets (t+2)(t-6)>=0, and the solution gives t=6, that is, (ab)min=6, and a=b=3

In the same way, if you find the minimum value of ab in the same way, can you get 9?

According to the fundamental inequality ab [a b 2].

a＋b＝ab≤[﹙a＋b﹚／2]²

i.e. a b 4 a b 0

The minimum value of a b 4a b is 4

A positive number is a mathematical term, and a number larger than 0 is called a positive number, and 0 itself is not a positive number. Positive numbers and negative numbers represent quantities with opposite meanings. Positive numbers are often preceded by a symbol "+" which can usually be omitted, and negative numbers are marked with a minus sign (equivalent to a minus sign)" and a positive number, such as 2, which represents the opposite of 2.

On the number axis, the positive numbers are on the right side of 0, and the earliest record of positive numbers is the ancient mathematical work "Nine Chapters of Arithmetic" in China.

(a+b)=ab<=((a+b)/2)^2a+b>=4

Yes and only if a=b=2 is the minimum.

a+b)min=4

ab-3=a+b

ab＝a＋b＋3≥2√ab ＋3

Limb Jane ab 2 ab 3 0

Solution: ab 3 or ab 1 (discarding the calendar pants) ab 9 if and only if a b, the equal sign holds

The minimum value of ab is 9 lifetimes.

ab=a+b+3

a+b=ab-3

a+b)^=a^b^-6ab+9

a^+b^=(ab-4)^-7

The most only small value to destroy the fiber for the fiber mountain imitation-7

If the positive real numbers a and b satisfy ab=2, then the minimum value of a+b is .

To get the minimum value of the sum of a and b, we use the arithmetic-geometric mean inequality (am-gm inequality). For the non-negative real numbers a and b, there is the following inequality: (a + b) 2 ab) is run on the regular knowledge ab = 2, and we need to find the minimum value of Qingwu a + b.

According to the am-gm inequality, we have: (a + b) 2 2 now solves a + b: a + b 2 2 Therefore, the minimum value of a + b is 2 2.

Untie. a,b>0

a+b=1(2/a+4/b)(a+b)

2+(2b/a)+(4a/b)+4

6+(2b/a)+(4a/b)

6+2√(2b/a)×(4a/b)

If and only if 2 a = 4 b, the equal sign holds.

i.e. 2a = b = 1 3

The minimum value of 2 a+4 b is: 6 + 4 2

Let a+b=m, then ab=m+3, and take a and b as the root to get x2mx+m+3=0, =m2

4（m+3）=m2

4m-12 0, and m 0, solution, m 6, a2b2(a+b)2

2ab=（m-1）2

7, when m=6, a2

b2 can achieve the minimum value of the hole chain is 18

So the answer is: 18 elimination.

ab-(a+b)=1

ab-a-b+1=2

a-1)(b-1)=2

a+b(a-1)+(b-1)+2

2+2 peeling [(a-1)(b-1)].

2+2 Guess 2

The minimum value of a+b is 2+2 2

Let A+ 4B t

a=t-4b

a+ b+ ab=8

t-4b)+b+b(t-4b) 84b 2-(t-3)b+8-t=0

t-3) 2-16(8-t)>next to grip position=0

t^2-10t-119>=0

t>=17 or t