Maximizing f(x,y) on the circle x^2 + y^2 = 12

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In summary, the problem involves maximizing the function f(x,y) = x^2y constrained by the circle x^2 + y^2 = 12. The attempt at a solution involved solving for lambda, but it could not be plugged in to solve for x and y as it is still a variable. By eliminating lambda from the two equations, a curve is obtained where the gradient of f is perpendicular to circles centered on the origin. The intersection of this curve with the given circle can then be found to determine the maximum value of f.
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dazedoutpinoy
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Homework Statement

Maximize the function f(x,y) = x^2y constrained by the circle x^2 + y^2 = 12





The attempt at a solution

I already went as far as solving lambda in my work; however, it's still a variable so I could not plug it into solve for x and y.

http://img.photobucket.com/albums/v407/dazedoutpinoy/Calculus001.jpg"
 
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(it took me a while to understand the meaning of (i) and (j), ...)

From the two equations you have writen, you can now eleminate lambda.
In this way, you will get a curve (two lines actually) where the gradient of f is perpendicular to circles centered on the origin.
(Df = lambda Dg means that as you know: the gradient of f should have no finite component along the given circle)

Then, you should simply find out the intersection of this curve with the particular circle you are targeting.
 

FAQ: Maximizing f(x,y) on the circle x^2 + y^2 = 12

What does it mean to "maximize the function"?

Maximizing the function refers to finding the highest possible value for a given mathematical function. This can be achieved by finding the maximum point on the graph or by using calculus to find the critical point where the function's derivative is equal to zero.

Why is maximizing the function important?

Maximizing the function is important because it helps us understand the behavior and characteristics of the function. It can also help us make decisions and optimize processes in fields such as economics, engineering, and science.

What are some common techniques for maximizing a function?

Some common techniques for maximizing a function include using calculus to find the critical point, using optimization algorithms such as gradient descent or genetic algorithms, and using mathematical tools such as Lagrange multipliers.

Can any function be maximized?

It depends on the function and the constraints placed on it. In some cases, a function may have a global maximum that can be found, while in other cases, it may have multiple local maxima or no maximum at all.

How do you know if a function has been maximized?

You can determine if a function has been maximized by finding the maximum point on the graph or by using mathematical techniques to prove that the function's derivative is equal to zero at the critical point. Additionally, you can compare the value of the function at the critical point to the values of the function at other points to see if it is the highest possible value.

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