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Question Number 126144 by Ndala last updated on 17/Dec/20

$$if:a+b=c$$ $$Provethat:a^{\frac{2}{3}}+b^{\frac{2}{3}}>c^{\frac{2}{3}}$$ $$.$$ $$\mathrm{H}elpme,please!$$

Answered by MJS_new last updated on 18/Dec/20

$$a^{2/3}+b^{2/3}>{(a+b)}^{2/3}$$ $${(a^{2/3}+b^{2/3})}^3>{(a+b)}^2$$ $$a^2+3a^{4/3}{b}^{2/3}+3a^{2/3}{b}^{4/3}+b^2>a^2+2ab+b^2$$ $$3a^{2/3}{b}^{2/3}(a^{2/3}+b^{2/3})>2ab$$ $$3(a^{2/3}+b^{2/3})>2a^{1/3}{b}^{1/3}$$ $$a^{2/3}-\frac{2}{3}{a}^{1/3}{b}^{1/3}+b^{2/3}>0$$ $$x^2-\frac{2}{3}xy+y^2>0\mathrm{true}\mathrm{for}x,y\in \mathbb{R}\mathrm{\setminus }\left\{0\right\}$$

Commented byNdala last updated on 21/Dec/20

$$\mathrm{Thanks}!$$

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