Max & Min Values of $S$ for $x_1^2 + x_2^2 = y_1^2 + y_2^2 = 2013$

• MHB
• anemone
In summary, the maximum value of $S$ is $4$ and the minimum value is $0$ when $x_1^2 + x_2^2 = y_1^2 + y_2^2 = 2013$.
anemone
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MHB
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Find the maximum and the minimum values of $S = (1 - x_1)(1 -y_1) + (1 - x_2)(1 - y_2)$ for real numbers $x_1, x_2, y_1,y_2$ with $x_1^2 + x_2^2 = y_1^2 + y_2^2 = 2013$.

Hi there,

To find the maximum and minimum values of $S$, we can use the method of Lagrange multipliers. This method involves finding the critical points of a function subject to some constraints.

In this case, our function is $S = (1 - x_1)(1 -y_1) + (1 - x_2)(1 - y_2)$ and our constraints are $x_1^2 + x_2^2 = y_1^2 + y_2^2 = 2013$.

Using Lagrange multipliers, we can set up the following equations:

$\frac{\partial S}{\partial x_1} = \lambda \frac{\partial (x_1^2 + x_2^2)}{\partial x_1}$
$\frac{\partial S}{\partial x_2} = \lambda \frac{\partial (x_1^2 + x_2^2)}{\partial x_2}$
$\frac{\partial S}{\partial y_1} = \lambda \frac{\partial (y_1^2 + y_2^2)}{\partial y_1}$
$\frac{\partial S}{\partial y_2} = \lambda \frac{\partial (y_1^2 + y_2^2)}{\partial y_2}$
$x_1^2 + x_2^2 = y_1^2 + y_2^2 = 2013$

Solving these equations, we get the following critical points:

$(x_1, x_2, y_1, y_2) = (\pm \sqrt{2013}, 0, 0, \pm \sqrt{2013})$ and $(x_1, x_2, y_1, y_2) = (0, \pm \sqrt{2013}, \pm \sqrt{2013}, 0)$.

1. What is the maximum and minimum value of S?

The maximum value of S is 2013, which occurs when x1 = y1 = 0 and x2 = y2 = √2013. The minimum value of S is 0, which occurs when x1 = x2 = y1 = y2 = 0.

2. How do you find the maximum and minimum value of S?

To find the maximum and minimum value of S, we can use the Pythagorean theorem and the fact that x1^2 + x2^2 = y1^2 + y2^2 = 2013. By substituting the values of x1, x2, y1, and y2, we can solve for S.

3. Can S be a negative value?

No, S cannot be a negative value. This is because S represents the difference between the squares of two numbers, and the square of any real number is always positive.

4. What happens if the equation is changed to x1^2 + x2^2 = y1^2 + y2^2 = k, where k is a positive integer?

The maximum value of S would still be k, but the minimum value of S would be 0. This is because the minimum value of S occurs when all variables are equal to 0, which is always possible when k is a positive integer.

5. Can the maximum and minimum values of S be the same?

Yes, the maximum and minimum values of S can be the same. This occurs when x1 = x2 = y1 = y2 = 0, which results in S = 0.

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