The sum of two positive numbers is 48. What is the smallest possible value of the sum of their squares?
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You are looking for a minimum of: f(x) = x^2 + (48 - x)^2 which can be re-written as: f(x) = x^2 + 2302 - 96x + x^2 First derivative is: f'(x) = 4x - 96 Solve for zero to find extreme(s): 0 = 4x - 96 4x = 96 x = 24 That it's minimum you can confirm calculating second derivative: f''(x) = 4 Since it is positive, it is indeed a minimum. The function f(x) thus has its minimal value for x = 24. Since the question asks for the function value, the answer is f(24) = 24^2 + 24^2 = 1152. Which is your answer.
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