Question 1005533
Find two numbers whose sum is 56 and whose product is as large as possible. [Hint: Let x and 56-x be the two positive numbers. Their product can be described by the function f(x)=x(56-x).]
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28*28 = 784, the largest possible product.
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x = 28 --> 28*28
Any other combination is (x-a)*(x+a) where a is and integer < 28
(x-a)*(x+a) = x^2 - a^2 which is obviously less than x^2
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The hard way:
f(x) = x*(56-x) = -x^2 + 56x
The max is the vertex at x = -b/2a = -56/-2 = 28
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Using calculus:
f(x) = -x^2 + 56x
f'(x) = -2x + 56 = 0
x = 28 (again)