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Question Number 144264 by loveineq last updated on 24/Jun/21

$$\mathrm{Let}a,b>0\mathrm{and}2a+b=3.\mathrm{Prove}\mathrm{the}\mathrm{followings}:$$ $$\left(1\right)\frac{2}{n}a(b+4)+3b^{\frac{1}{n}}⩽\frac{10+3n}{n},\mathrm{\forall }n\in {\mathbb{N}}^{+}⩾1.$$ $$\left(2\right)2na(b+4)+3b^n⩾10n+3,\mathrm{\forall }n\in {\mathbb{N}}^{+}⩾2.$$ $$$$

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