find-the-minimum-value-a-a-2-8bc-b-b-2-8ac-c-c-2-8ab-

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

f i n d t h e m i n i m u m v a l u e

\frac{a}{\sqrt{a^{2} + 8 b c}} + \frac{b}{\sqrt{b^{2} + 8 a c}} + \frac{c}{\sqrt{c^{2} + 8 a b}}

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

t r y i n g t o s o l v e \dots

\frac{a^{2} + 8 b c}{2} ⩾ \sqrt{a^{2} \times 8 b c}

a^{2} + 8 b c ⩾ \sqrt{32 a b c \times a}

\sqrt{a^{2} + 8 b c} ⩾ {(32 a b c \times a)}^{\frac{1}{4}}

\frac{1}{\sqrt{a^{2} + 8 b c}} ⩽ \frac{1}{{(32 a b c \times a)}^{\frac{1}{4}}}

\frac{a}{\sqrt{a^{2} + 8 b c}} ⩽ \frac{a}{{(32 a b c \times a)}^{\frac{1}{4}}}

\frac{a}{\sqrt{a^{2} + 8 b c}} + \frac{b}{\sqrt{b^{2} + 8 a c}} + \frac{c}{\sqrt{c^{2} + 8 a b}} ⩽ \frac{a^{\frac{3}{4}} + b^{\frac{3}{4}} + c^{\frac{3}{4}}}{{(32 a b c)}^{\frac{1}{4}}}

\frac{a^{\frac{3}{4}} + b^{\frac{3}{4}} + c^{\frac{3}{4}}}{3} \times \frac{1}{{(32 a b c)}^{\frac{1}{4}}} ⩾ \frac{1}{{(32 a b c)}^{\frac{1}{4}}} \times {(a^{\frac{3}{4}} b^{\frac{3}{4}} c^{\frac{3}{4}})}^{\frac{1}{3}}

= \frac{1}{{(32)}^{\frac{1}{4}}}