Can You Find the Greatest Value Using Only AM-GM Inequality?

[utkarshakash]

Homework Statement

Find the greatest value of $x^3y^4$ if 2x+3y=7 and x>=0,y>=0

Homework Equations

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 :(

 

[Mark44]

Homework Statement

Find the greatest value of $x^3y^4$ if 2x+3y=7 and x>=0,y>=0

Homework Equations

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 :(

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.

 

[Ray Vickson]

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.

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

 

[utkarshakash]

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

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.