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# x+y=a, x*y= the largest possible value

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If x+y=a, where a, x, and y are positive, real numbers, then will the largest number that x*y can equal be if x=y? For example x+y=16. Then the largest their product can be will if x=y, so 8+8=16 and 8*8=64. Thus 64 is larger than 9*7 which also satisfies x+y=16 I hope this makes sense, thanks!

Jan 24, 2019

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$$x+y=a\\ y = a-x\\ x y = x(a-x) =\\ -(x^2 - a x)= \\ -\left(x^2 - ax +\dfrac{a^2}{4}-\dfrac{a^2}{4}\right) = \\ -\left(x-\dfrac a 2\right)^2 + \dfrac{a^2}{4}$$

$$xy \text{ will be a maximum of }\dfrac{a^2}{2} \\ \text{when }x = \dfrac a 2 \text{ i.e. when}\\ x = y = \dfrac a 2$$

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Jan 24, 2019