Finding the maximum of a product

[Mr Davis 97]

Homework Statement

If $A,B,C,D$ are positive numbers such that $A+2B+3C+4D = 8$, then what is the maximum value of $ABCD$?

Homework Equations

The Attempt at a Solution

From AM-GM, I know that the maximum is $2/3$. However, I want to know exactly what A,B,C, and D must be to take on this maximum. From AM-GM, I know that it must be the case that A=2B=3C=4D for equality to occur, but I'm not sure how I can use this explicitly, as a four-way equality is unwieldy.

 

[fresh_42]

What is the problem? If you have $A=2B=3C=4D$ then $8=A+2B+3C+4D=4A=(4\cdot 2)B=(4\cdot 3)C=(4\cdot 4)D$.

 

[Ray Vickson]

Homework Statement

If $A,B,C,D$ are positive numbers such that $A+2B+3C+4D = 8$, then what is the maximum value of $ABCD$?

Homework Equations

The Attempt at a Solution

From AM-GM, I know that the maximum is $2/3$. However, I want to know exactly what A,B,C, and D must be to take on this maximum. From AM-GM, I know that it must be the case that A=2B=3C=4D for equality to occur, but I'm not sure how I can use this explicitly, as a four-way equality is unwieldy.

This is a standard constrained multivariate optimization problem, and can be solved using a Lagrange multiplier method, for example.