Maxima and Minima

Calculus Level 3

If x y z = 8 xyz=8 , find the value of x + y + z x+y+z for which 5 x y z x + 2 y + 4 z \frac { 5xyz }{ x+2y+4z } is maximum.


The answer is 7.

Swapnil Roge by Swapnil Roge , 23, India

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1 solution

Rishabh Jain
Jan 1, 2016

Just a application of Cauchy-Schwarz , given function is maximum, when it's denominator is a minimum (since denominator is constant) and it occurs when
x/1=y/2=z/4=(x+y+z)/7=k(let) , using abc=8, we get k=1. Hence a+b+c=7k=7

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Can you find individual values of x , y , z x,y,z ?

Swapnil Roge - 5 years, 5 months ago

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1,2,4 respectively..

Rishabh Jain - 5 years, 5 months ago

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This is also solvable by Lagrange's Multipliers Method

Swapnil Roge - 5 years, 5 months ago

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