What is the maximum value of x2y3x^2 y^3 for positive numbers xx and yy satisfying x+y=2x+y=2?

Note by Abhishek Alva
2 years, 11 months ago

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Is it 3456/3125. ?

naitik sanghavi - 2 years, 11 months ago

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please write the solution

abhishek alva - 2 years, 11 months ago

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Am i correct?

naitik sanghavi - 2 years, 11 months ago

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@Naitik Sanghavi yes u are write the solution

abhishek alva - 2 years, 11 months ago

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@Abhishek Alva Apply Am gm on x,x,2y/3,2y/3,2y/3.

naitik sanghavi - 2 years, 11 months ago

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@Naitik Sanghavi a/(a^3+b^2+c )+b/(b^3+c^2+a)+c/(c^3+a^2+b) if a+b+c=3 determine the largest value of the equation given that a b c are positive real numbers

abhishek alva - 2 years, 11 months ago

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@Abhishek Alva i not getting this problem

abhishek alva - 2 years, 11 months ago

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I got the answer to be 676263125\dfrac{676\sqrt{26}}{3125} which is very close to your answer.First we get the max. value of xyxy to be 11 and x2+y2x^2+y^2 to be 22 by AMGMAM-GM inequality.Now we must find the max value of 2x+3y.2x+3y.We have an inequality which states that ax+bya2+b2×x2+y2.ax+by\leq \sqrt{a^2+b^2}\times\sqrt{x^2+y^2}.Well I dont know the proof for this inequality.Hence, 2x+3y22+32×2=26.2x+3y\leq\sqrt{2^2+3^2}\times\sqrt 2=\sqrt{26}. Next we apply AM-GM inequality for 2x+3y.2x+3y.We split the terms and apply the inequality.x+x+y+y+y5x2y352x+3y5x2y35..\dfrac{x+x+y+y+y}{5}\geq\sqrt[5]{x^2y^3}\Rightarrow \dfrac{2x+3y}{5}\geq\sqrt[5]{x^2y^3}.Therefore,265x2y35.\dfrac{\sqrt{26}}{5}\geq\sqrt[5]{x^2y^3}.Now raise both sides by power of 5.5.We get the maximum value of x2y3x^2y^3 to be 676263125\boxed{\dfrac{676\sqrt{26}}{3125}}..

Ayush G Rai - 2 years, 11 months ago

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When you applied am gm on x,x,y,y,y the equality holds when x=y which implies that max value of x²y³ is 1.I think this was ur mistake @Ayush Rai.

naitik sanghavi - 2 years, 11 months ago

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But it is very close to the answer.Can u point the error?

Ayush G Rai - 2 years, 11 months ago

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@Ayush G Rai I already did iny other comment

naitik sanghavi - 2 years, 11 months ago

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i guess u used the cauchy schwarz inequality

abhishek alva - 2 years, 11 months ago

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