# Find the greatest value

1. Nov 20, 2012

### utkarshakash

1. The problem statement, all variables and given/known data
Find the greatest value of $x^3y^4$ if 2x+3y=7 and x>=0,y>=0

2. Relevant equations

3. The attempt at a solution
Let the 7 numbers be (x/3) 3 times and (y/4) 4 times
Using AM GM inequality
$\dfrac{ 3.\frac{x}{3} + 4.\frac{y}{4}}{7} \geq \left[ \left( \frac{x}{3}\right)^3 . \left( \frac{y}{4}\right)^4\right]^{1/7} \\ \left( \dfrac{x+y}{7} \right)^7 \times 3^3.4^4 \geq x^3y^4$
But I'm stuck here :(

2. Nov 20, 2012

### Staff: Mentor

It's not clear in your problem statement, but I believe the restriction of x ≥ 0, y ≥ 0 applies to the linear equation, 2x + 3y = 7.

Sketch a graph of the portion of this line that lies in the first quadrant. Then solve this equation for one of its variables to substitute into x3y4 to make this a function of one variable.

3. Nov 20, 2012

### Ray Vickson

Alternatively, you can use the Lagrange multiplier method. Or, you can recognize this as a so-called "Geometric Programming Problem" and use methods devised for those types of problems.

RGV

4. Nov 21, 2012

### utkarshakash

Thanks. It solved my problem. Though I did not know Lagrange multiplier method but a little GOOGLing around helped me learn this method. But I'm required to solve this using only the A.M. G.M. inequality. Btw thanks for introducing this method to me. It will be really helpful in solving complicated problems.