Let x and y be nonnegative real numbers. if x^2+5y^2=30, what is the max value of xy?

gaberiellz May 4, 2024

#1**-1 **

Absolutely, I’ve been improving my problem-solving abilities in maximizing functions under constraints. Let's maximize the expression: xy

subject to the constraint: 30x2+305y2=1

We can solve this problem using Lagrange multipliers. Here's how we can approach this problem:

Introduce Lagrange Multiplier: We'll introduce a Lagrange multiplier, λ, to incorporate the constraint into the objective function. The new function to maximize will be:

L(x,y,λ)=xy+λ(30x2+305y2−1)

Take Partial Derivatives: We will take partial derivatives of L with respect to x, y, and λ, and set them equal to zero. This will give us a system of equations to solve for x, y, and λ.

∂x∂L=y+λ(302x)=0

∂y∂L=x+λ(3010y)=0

∂λ∂L=30x2+305y2−1=0

Solve the System of Equations: From the first equation, we get:

y=−λ(302x)=−15λx

Substitute this into the second equation:

x−λ(3010⋅−15λx)=0

x+452λ2x=0

x(4545+2λ2)=0

Since x is a nonnegative real number, we can ignore the case where x = 0. Therefore:

4545+2λ2=0

2λ2=−45

This equation has no real solutions for λ. However, we can notice that the constraint equation is already satisfied regardless of the value of x and y. This means that any values of x and y that satisfy the constraint will also satisfy the condition where the partial derivatives are zero.

Find Maximum Value: Since the constraint forces x2+5y2=30, the largest possible value of xy will occur when x and y are as close in value as possible, subject to the constraint. In a 2D plane, this would correspond to a point on the ellipse where x and y have the same absolute value.

Looking at the constraint equation, we see that when x2=18 and y2=12 (or vice versa), the condition is satisfied. So, one possible solution is:

x= sqrt(18) = 3 sqrt(2)

y= sqrt(12) = 2 sqrt(3)

Therefore, the maximum value of xy is:

xy = 3*sqrt(2)*2*sqrt(3) = 6*sqrt(6)

In conclusion, the maximum value of xy subject to the constraint is 6*sqrt(6).

Akhain1 May 4, 2024

#3**+2 **

Hey Akhain,

Can you please explain to me how you found the contraint 30x2+305y2=1?

My puny brain is not exactly sure I understand that part....

Thanks!

NotThatSmart
May 4, 2024