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anemone
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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 $x_1^2 + x_2^2 = y_1^2 + y_2^2 = 2013$.
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.
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.
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.
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.
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.