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Question Number 55326 by tm888 last updated on 21/Feb/19

$$findtheminimumvalue$$$$\frac{a}{\sqrt{a^2+8bc}}+\frac{b}{\sqrt{b^2+8ac}}+\frac{c}{\sqrt{c^2+8ab}}$$

Answered by tanmay.chaudhury50@gmail.com last updated on 21/Feb/19

$$tryingtosolve...$$$$\frac{a^2+8bc}{2}⩾\sqrt{a^2\times 8bc}$$$$a^2+8bc⩾\sqrt{32abc\times a}$$$$\sqrt{a^2+8bc}⩾{\left(32abc\times a\right)}^{\frac{1}{4}}$$$$\frac{1}{\sqrt{a^2+8bc}}⩽\frac{1}{{\left(32abc\times a\right)}^{\frac{1}{4}}}$$$$\frac{a}{\sqrt{a^2+8bc}}⩽\frac{a}{{\left(32abc\times a\right)}^{\frac{1}{4}}}$$$$\frac{a}{\sqrt{a^2+8bc}}+\frac{b}{\sqrt{b^2+8ac}}+\frac{c}{\sqrt{c^2+8ab}}⩽\frac{a^{\frac{3}{4}}+b^{\frac{3}{4}}+c^{\frac{3}{4}}}{{\left(32abc\right)}^{\frac{1}{4}}}$$$$\frac{a^{\frac{3}{4}}+b^{\frac{3}{4}}+c^{\frac{3}{4}}}{3}\times \frac{1}{{\left(32abc\right)}^{\frac{1}{4}}}⩾\frac{1}{{\left(32abc\right)}^{\frac{1}{4}}}\times {\left(a^{\frac{3}{4}}{b}^{\frac{3}{4}}{c}^{\frac{3}{4}}\right)}^{\frac{1}{3}}$$$$=\frac{1}{{\left(32\right)}^{\frac{1}{4}}}$$$$$$$$$$

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