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Nonnegative real numbers a and b satisfy sqrt(a)-sqrt(b)=20. Find the maximum value of a-5b? Thanks in advance. 

 Jul 26, 2019
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\(\sqrt{a}-\sqrt{b}=20\\ a = (20+\sqrt{b})^2\\ a-5b = (20+\sqrt{b})^2-5b\\ \dfrac{d}{db} (20+\sqrt{b})^2-5b = \\ 2(20+\sqrt{b})\cdot \dfrac{1}{2\sqrt{b}} - 5\\ \text{and we set this equal to 0}\)

 

\(\dfrac{20}{\sqrt{b}}+1 = 5\\ \sqrt{b}=5\\ b=25\\ \sqrt{a}-5=20\\ a=625\)

 

\(\text{Max value of $a-5b=625-5(25)=500$}\)

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 Jul 26, 2019

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