Find the relative maxima or minima in Exercises 1-10 Maximum of f(x, y)=x y^2, subject to x+2 y=

Find the relative maxima or minima in Exercises 1-10 Maximum...

Key Concepts

Objective Function

The objective function is the specific function that is being optimized, either maximized or minimized, in an optimization problem. It represents the measure of performance or cost that one aims to improve or reduce.

Constrained Optimization

Constrained optimization involves finding the maximum or minimum of an objective function while fulfilling one or more restrictions or constraints. It is used across various fields to determine the best outcome within a limited set of conditions.

Lagrange Multipliers

The Lagrange multiplier method is a strategy for solving constrained optimization problems. It transforms a problem with constraints into an equation system where the gradients of the objective function and the constraint are proportional, thereby enabling the identification of potential optimal points.

Constraint Equation

A constraint equation is an equation that must be satisfied by any solution in a constrained optimization problem. It limits the possible values of the variables involved, ensuring that the solution adheres to the established conditions.

Transcript

00:01 Okay, so to start, our function that we're trying to maximize is xy squared.

00:12 Let's note that right now.

00:18 Maximize this function, and then we have this constraint.

00:22 The constraint equation is x plus 2y equals 15, and then that gives our constrained function of x plus 2y minus 15 so then our function of function of three variables to use the lagrange multiple supplier method is f of x y minus lambda times g of x y so in general this is once you have your function that you're trying to maximize and your constraint equation, you create a third function, and then what we want to do is we want to solve this system of three equations and three unknowns.

01:15 So the partial of this function with respect to x equals zero, and then the partial of that function with respect to y equals zero, and the partial of that function with respect to land.

01:32 Equals 0.

01:34 So i have three equations, three unknowns.

01:37 So let's go ahead and write this function out.

01:44 So it's x y squared minus lambda x minus 2 lambda y plus 15 lambda.

01:59 And don't forget that this negative needs to be distributed to all those pieces and now we start taking our partial derivatives we're going to end up with quite a few equations and i'm going to shorten this to just partial with respect to x is going to be y squared minus lambda and we're going to want to that to equal zero.

02:28 Second equation partial with respect to y, 2xy minus 2 lambda, equal 0, and then our third equation is the partial with respect to lambda, negative x, minus 2y plus 15 equals 0.

02:57 All right, so the first equation gives us, so y squared equals lambda, so y is plus or minus the square root of lambda.

03:12 And then the second equation and the first, well, let's do the second equation gives us, let's solve that for x.

03:22 So we get 2xy minus or sorry equals 2 lambda.

03:29 So we get x equals lambda over y.

03:35 So and y was plus or minus the square root of lambda...

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