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Author: Tinku Tara

Question-200894

Question Number 200894 by sonukgindia last updated on 26/Nov/23 Answered by Frix last updated on 26/Nov/23 $$=\mathrm{4}\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\frac{{dx}}{\mathrm{1}+\mathrm{4sin}^{\mathrm{2}} \:{x}}\:\overset{{t}=\mathrm{tan}\:{x}} {=} \\ $$$$=\mathrm{4}\underset{\mathrm{0}} {\overset{\infty} {\int}}\frac{{dt}}{\mathrm{5}{t}^{\mathrm{2}}…

Question-200942

Question Number 200942 by Mingma last updated on 26/Nov/23 Answered by witcher3 last updated on 26/Nov/23 $$\frac{\mathrm{1}}{\mathrm{a}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{b}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{c}^{\mathrm{2}} }\geqslant\mathrm{3}\left(\mathrm{abc}\right)^{−\frac{\mathrm{2}}{\mathrm{3}}} \:\:\mathrm{AM}−\mathrm{GM} \\ $$$$\mathrm{ab}+\mathrm{bc}+\mathrm{ac}\geqslant\Leftrightarrow \\ $$$$\mathrm{abc}\left(\frac{\mathrm{1}}{\mathrm{a}}+\frac{\mathrm{1}}{\mathrm{b}}+\frac{\mathrm{1}}{\mathrm{c}}\right)\geqslant\mathrm{1}…