The sum of two non negative numbers is 20. Find the sum of the squares that is
A.) As large as possible
B.) As small as possible
That is two seperate questions, not mutiple choice... I am in the unit of curve sketching, optimization, physimatics and stuff like that
A.) As large as possible
B.) As small as possible
That is two seperate questions, not mutiple choice... I am in the unit of curve sketching, optimization, physimatics and stuff like that
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Call the numbers x, and y. The x ≥ 0, y ≥ 0 and x + y = 20. You can write y = 20 - x. So the task is to optimize
x² + y² = x² + (20-x)² = S(x).
S(x) = 2x²-40x + 400, 0 ≤ x ≤ 20.
The extreme value theorem says that S takes its maximum and minimum on this closed interval.
S'(x) = 4x - 40 ==> S'(x) = 0 if x = 10.
S(0) = 400, S(10) = 200, S(20) = 400.
The maximum value is 400 and the minimum possible value is 200.
x² + y² = x² + (20-x)² = S(x).
S(x) = 2x²-40x + 400, 0 ≤ x ≤ 20.
The extreme value theorem says that S takes its maximum and minimum on this closed interval.
S'(x) = 4x - 40 ==> S'(x) = 0 if x = 10.
S(0) = 400, S(10) = 200, S(20) = 400.
The maximum value is 400 and the minimum possible value is 200.
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The question says "Find the sum of the squares that is..." The 200 and 400 are those sums. The x and y values are there too. To get max, take x = 0, y = 20 or x = 20, y = 0. To get the min take x = y = 10. It's all there in my solution.
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