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

#Algebra

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Is it 3456/3125. ?

naitik sanghavi - 4 years, 8 months ago

please write the solution

abhishek alva - 4 years, 8 months ago

Am i correct?

naitik sanghavi - 4 years, 8 months ago

@Naitik Sanghavi – yes u are write the solution

abhishek alva - 4 years, 8 months ago

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

naitik sanghavi - 4 years, 8 months ago

@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 - 4 years, 8 months ago

@Abhishek Alva – i not getting this problem

abhishek alva - 4 years, 8 months ago

I got the answer to be $\dfrac{676\sqrt{26}}{3125}$ which is very close to your answer.First we get the max. value of $xy$ to be $1$ and $x^2+y^2$ to be $2$ by $AM-GM$ inequality.Now we must find the max value of $2x+3y.$ We have an inequality which states that $ax+by\leq \sqrt{a^2+b^2}\times\sqrt{x^2+y^2}.$ Well I dont know the proof for this inequality.Hence, $2x+3y\leq\sqrt{2^2+3^2}\times\sqrt 2=\sqrt{26}.$ Next we apply AM-GM inequality for $2x+3y.$ We split the terms and apply the inequality $.\dfrac{x+x+y+y+y}{5}\geq\sqrt[5]{x^2y^3}\Rightarrow \dfrac{2x+3y}{5}\geq\sqrt[5]{x^2y^3}.$ Therefore, $\dfrac{\sqrt{26}}{5}\geq\sqrt[5]{x^2y^3}.$ Now raise both sides by power of $5.$ We get the maximum value of $x^2y^3$ to be $\boxed{\dfrac{676\sqrt{26}}{3125}}$ $.$

Ayush G Rai - 4 years, 8 months ago

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 - 4 years, 8 months ago

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

Ayush G Rai - 4 years, 8 months ago

@Ayush G Rai – I already did iny other comment

naitik sanghavi - 4 years, 8 months ago

i guess u used the cauchy schwarz inequality

abhishek alva - 4 years, 8 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$