The Lagrange Multipliers Needed to Find the Maximum and Minimum $f ( x , y ) =$

question 43

Essay

The Lagrange multipliers needed to find the maximum and minimum values of $f ( x , y ) =$ 8xy on the circle $x ^ { 2 } + y ^ { 2 } = 25$ is $\lambda = \pm 4$ , where $\nabla f = \lambda \nabla g$ .Estimate the maximum and minimum values of 8xy subject to the constraint $x ^ { 2 } + y ^ { 2 } = 30.25$ .

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Definitions:

Producing

The action of making or manufacturing from components or raw materials.

Producer Surplus

The discrepancy between the price at which producers are ready to offer a good and the actual amount they get for it.

Market Equilibrium

A situation in which market supply equals market demand, leading to stable prices where the quantity supplied is equal to the quantity demanded.

Consumer Surplus

The disparity between what consumers are prepared and capable of spending for a product or service and the actual amount they end up paying.