Find the relative maxima or minima in Exercises 1-10 Maximum of f(x, y)=3 x y , subject to x+2 y

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

Key Concepts

Lagrange Multipliers

The method of Lagrange multipliers is a strategy used to find the local maxima and minima of a function subject to equality constraints. It introduces additional variables, called Lagrange multipliers, to incorporate the effects of the constraints into the optimization process, allowing the simultaneous resolution of the function and its constraints.

Critical Point Analysis

Critical point analysis in multivariable calculus involves identifying points where the gradient of the function is zero or undefined. When dealing with constrained optimization problems, critical points are typically found by setting up conditions where the gradients of the function and the constraints align, leading to potential relative extrema.

Constrained Optimization

Constrained optimization involves finding the extrema (maximum or minimum values) of a function while satisfying one or more constraints. In many practical problems, the variables are linked by constraints, and the optimal solution must comply with these restrictions. This concept is foundational in areas such as economics, engineering, and operations research.

Transcript

00:01 Okay, so for this question we have, i'm going to find the relative maxima or minima, and the function that we're trying to maximize slash minimize is f of x y equals 3xy and then we have this constraint function.

00:26 So i'm actually it's a constraint equation and we have to convert that into a function.

00:39 So we convert that into a function by setting it equal to zero or the function that we are interested in.

00:48 Alright, so this is our constraint.

00:55 And this is what we want to find the max, relative max or min of.

01:05 Now, the method of lagrange multiplies.

01:07 Multipliers, so just in general, is we take this function that we're trying to optimize, and we take this constraint function, and we build another function, and it's a function of three variables.

01:31 X, y, and lambda, this is the lagrange multiplier, and this is defined as f of x, y, so this function that we're trying to maximize minus lambda times g of xy our constraint function and then once we have that then we have to solve partial with respect to x equal zero partial with respect to y equal zero partial with respect to y equal zero with respect to lambda equal zero.

02:16 So we'll end up with three equations and three unknowns, and the unknowns are x, y, and lambda.

02:25 So in our particular case, so this is maybe just like, in general, this is what we want to do.

02:37 So for our particular problem, our function of three variables is, 3xy minus lambda x minus 2 lambda y plus 32.

02:58 Lambda make sure that this negative is being distributed to all pieces of the g of x function.

03:05 And now f, the partial with respect to x, i'll just denote it like that, is 3y minus lambda, and the partial with respect to y is 3x.

03:20 Minus 2 lambda and the partial with respect to lambda is negative x minus 2y plus 32 and we want to solve we want all of these equal to 0 and we want to solve this system of three equations and three unknowns now if you know any linear algebra you can just set up a matrix and do that if you don't know linear algebra we can just do this by hand so let's call that 1, equation 2, and equation 3.

03:56 So equation 1 gives us that y equals 1 3rd lambda.

04:06 Equation 2 gives us that x equals 2 thirds lambda.

04:17 And then what we're trying to do basically is we're trying to combine these three equations to get one of them to have just one variable...

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