Two high school mathematical inequality questions 1. The known real number A.B.C satisfies a + B + C + D = 3. A ^ + 2B ^ + 3C ^ + 6D ^ = 5 What are the maximum and minimum values of a? 2. If a and B belong to R +, and ab - (a + b) = 1, what is the minimum value of a + B? Thank you very much

Two high school mathematical inequality questions 1. The known real number A.B.C satisfies a + B + C + D = 3. A ^ + 2B ^ + 3C ^ + 6D ^ = 5 What are the maximum and minimum values of a? 2. If a and B belong to R +, and ab - (a + b) = 1, what is the minimum value of a + B? Thank you very much

1. By Cauchy inequality (2B ^ 2 + 3C ^ 2 + 6D ^ 2) (1 / 2 + 1 / 3 + 1 / 6) > = (B + C + D) ^ 2
So (5-a ^ 2) > = (3-A) ^ 2
a^2-3a+2

It is known that ∠ xoy = 60 ° m is a point in ∠ xoy, the distance from it to ox is Ma = 2, and the distance from it to oy is MB = 11
Imagine for yourself,

If we extend the intersection of MB and ox to C, then ∠ OCB = 30 ° because Ma = 2, so MC = 4, so CB = cm + MB = 4 + 11 = 15. In the RT triangle CBO, Tan ∠ xoy = be / ob = √ 3, so ob = 5 √ 3. In the RT triangle MBO, MB = 11, OB = 5 √ 3, OM = 13 can be obtained from the Pythagorean theorem

It is known that ∠ xoy = 60 & ordm; m is a point in ∠ xoy, the distance from it to edge ox Ma = 2, and the distance from it to oy MB = 11 to find the OM length

Prolonging the intersection of AM and ob at point P
∠APO=30°
MP=2MB=22
AP=22+2=24
OA = 8 change number 3
Square of OM = square of Ma + square of OA
OM=14