Question

Use Lagrange multipliers to find three positive numbers whose sum is 210 and whose product is maximum. (Enter your answers as a comma-separated list.)

Answer 1

Answer: the three positive numbers are; 70, 70, 70

Explanation:

Given that sum is equal = 210

Lets ( x, y, z ) be the three positive numbers

such that

x + y + z = 210

what is the maximum of xyz

take f(x,y,z) = xyz

Q(x,y,z) = 0

x + y + z -210 = 0

consider the function

F(x,y,z) = f(x,y,z) + λQ(x,y,z)

F = xyz + λ(x+y+z-210)

dF/dx = 0 ⇒ yz + λ(1) = 0 ⇒ λ = -yz ..............equ(1)

dF/dy = 0 ⇒ xz + λ(1) = 0 ⇒ λ = -xz.................equ(2)

dF/dz = 0 ⇒ xy + λ(1) = 0 ⇒ λ = -xy...............equ(3)

Now

equ(1)/equ(2) ⇒ λ/λ = -yz/-xz ⇒ x = +y

equ(1)/equ(3) ⇒ λ/λ = - yz/-xy ⇒ x = +z

⇒ y = z = x

by substitution

x + y + z = 210

x + x + x = 210

3x = 210

x = 210/3 = 70

∴ x, y, z = 70, 70 ,70

MAXIMUM

∛xyz = 70 { when x = y = z = 70}