2 var inquality

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sqing

#1 h Mar 4, 2025, 12:46 PM

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Let $a,b>0$ and $3a+4b=a^3b^2.$ Prove that
$2a+b+\dfrac{2}{a}+\dfrac{3}{b}\geq \frac{11}{\sqrt2}$ $a+\dfrac{2}{a}+\dfrac{3}{b}\geq 4\sqrt[4]{\frac23}$ $\dfrac{2}{a}+\dfrac{3}{b}\geq 2\sqrt[4]3$ $3a+\dfrac{2}{a}+\dfrac{3}{b}\geq \sqrt[4]{354+66\sqrt{33}}$

This post has been edited 2 times. Last edited by sqing, Mar 4, 2025, 12:58 PM

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sqing

#2 h Mar 10, 2025, 8:38 AM

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Let $a,b>0$ and $\dfrac{1}{a}+\dfrac{2}{b}\leq 2 .$ Prove that
$a+b \leq a^3b^2$ Let $a,b>0$ and $\dfrac{3}{a}+\dfrac{2}{b}\leq 2\sqrt 3 .$ Prove that
$a+3b \leq a^3b^2$ Let $a,b>0$ and $\dfrac{1}{a}+\dfrac{8}{b}\leq 2\sqrt 2 .$ Prove that
$4a+b \leq a^3b^2$

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sqing

#3 h Mar 19, 2025, 5:00 AM

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Let $a,b>0$ and $3a+4b=a^3b^2.$ Prove that
$2a+b+ka^2b +\frac{2}{a}+\frac{3}{b}\geq \frac{6k+11}{\sqrt2}$ Where $k\geq 0.$
$2a+b+a^2b+\frac{2}{a}+\frac{3}{b} \geq \frac{17}{\sqrt2}$

This post has been edited 1 time. Last edited by sqing, Mar 19, 2025, 5:03 AM

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