Question

find the maximum and mimimum value of n(XUY) and n X(XnY), where n(x)=15 , n(y)=20

Answer 1

Step-by-step explanation:

Suppose that x, y, z . . . w are n positive variables and that c is a constant, then If x + y + z + ... + w = c, the value of xyz ... w is greatest when x = y = z = ... = w = c/n.

a + b + c = 12 , find maximum value of a.b.c

Ans : 64

a + b + c =19, find maximum value of a.b.c

Ans 19^3/27

a + b + c = 7 ; find Maximum value of a^2 * b^3 * c^4

Method 1:

a/2 + a/2 + b/3 + b/3 +b/3 + c/4 + c/4 + c/4 + c/4 = 7

now a/2 = a/2 = b/3 = b/3 = b/3 = c/4 = c/4 = c/4 = c/4 = 7/9

===> a/2 × a/2 × b/3× b/3 × b/3 × c/4 × c/4 × c/4 × c/4 = (7/9)^9

===> a^2. b^3. c^4 = 2^2 × 3^3 ×4^4 × (7/9)^9

Method 2 :

AM ≥ GM

a/2 + a/2 + b/3 + b/3 +b/3 + c/4 + c/4 + c/4 + c/4 = 7

{a/2 + a/2 + b/3 + b/3 +b/3 + c/4 + c/4 + c/4 + c/4 } /9 = { a/2 × a/2 × b/3× b/3 × b/3 × c/4 × c/4 × c/4 × c/4 }^1/9

Now on simplification we will get a^2 * b^3 * c^4 ≤ 2^2 × 3^3 ×4^4 × (7/9)^9