Suppose S = x + 2y is an objective function subject to th...

Question

Suppose S = x + 2y is an objective function subject to the constraint xy = 50, for x > 0 and y > 0.

b. Find the absolute minimum value of S subject to the given constraint.

Accepted Answer

Welcome back, everyone. In this problem, given the objective function Z equals 4 X + 3Y with the constraint that XY equals 72 for X and Y greater than 0, determine the absolute minimum value of Z. A says it's 24 route 6, B12 route 6, C 36 Route 6, and D 48 Route 6. Now, if we're going to figure out the absolute minimum value of Z, some type of differentiation is gonna be involved because we'll need to find the critical points. We need to test if they are maximum or minimum, and we can't do that without the 1st and 2nd derivatives of our function. So if we're going to differentiate it, we need to express it in terms of one variable. So let's express Z. OK, express Z in terms of in terms of X, sorry. To eventually be able to differentiate Z with respect to X. Now, we know Z equals 4 X + 3Y, but we also know that XY equals 72. So from this we can tell that Y. Is gonna be equal to 72 divided by X and if we substitute that into our objective function for Z, then that means Z is gonna be equal to 4 X plus 3 multiplied by 72 divided by X. Now if we simplify that, OK, we can write it as 4 X plus 216 divided by X. Now to find the critical points, we can differentiate Z of X with respect to X, OK? So now, this implies then that the derivative of Z of X, OK? And of course we're saying that this is Z as a function of X. So let's just specify that here. The derivative of ZF X. is going to be equal to 4 minus 216 divided by X2, OK? And now at the critical points of our function. Are the critical points. And that means our first derivative is gonna be equal to 0, OK? So, 4 minus 216 divided by X2 equals 0. If that's the case, that means 216 divided by X2 equals 4, and thus X2 equals 216 divided by 4 or 54. Therefore, that means X is going to be equal to route 54 or simplified 3 route 6. Now that we have a value for X, we can find a corresponding value of Y using the constraint equation. Remember we defined Y as 72 divided by X. That means then that Y is gonna be equal to 72. Divided by 3 route 6. Which equals 4 route 6, OK? It's gonna be 4 route 6. Know that we have Y and we have a value of X, let's verify that these values of X and Y are the absolute minimum, are at the points rather at the absolute minimum. And to do that, we can check the second derivative. So no, Verifying if these are the absolute minimum. X and Y are the absolute minimum. OK. We know that at the absolute minimum, OK, the second derivative is going to be greater than 0, OK? It's, it's supposed to be a positive point. So let's check that. Well, to find a second derivative. Means we're going to differentiate the first derivative. So you're gonna be finding the derivative, OK, of 4 minus 216 divided by X2, and the derivative is 432 divided by X cubed, OK? So now, at X equals 3 root 6. Then that means a 2nd derivative. It's gonna be equal to 432, divided by 3 roots 6 cubed. That's 432, OK, divided by 162, Route 6. And now we know that this value is gonna be positive, it's gonna be greater than 0. And since the second derivative is positive, that means the critical point, or no, we've verified that the critical point corresponds. To a minimum, OK. It's very important that we have to verify that. So now we know that it's a minimum, we can find, of course, the absolute minimum value of Z. So now substituting those values of X and Y into the objective function, OK. Substitute substituting. X and Y into the minimum function. Or into the objective function or rather. OK. Then now we know that Z will be equal to 4 X, so that's 4 multiplied by 3 root 6 plus 3 Y 3 multiplied by 4 root 6, OK, which is going to be equal to 2 multiplied by 12 route 6 or 24 route 6. Thus, the absolute minimum value of Z is 24 route 6, OK? If we go back to our answer choices, that means A is the correct answer. Thanks for watching everyone. I hope this video helped.

Suppose S = x + 2y is an objective function subject to the constraint xy = 50, for x > 0 and y > 0.
b. Find the absolute minimum value of S subject to the given constraint.

Key Concepts

Objective Function

Constraints

Lagrange Multipliers

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