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Show-that-for-all-a-b-c-d-R-with-a-b-c-d-0-1-ab-cd-1-4-a-2-b-2-c-2-d-2-2-abcd-1-4-1-4-a-b-c-d-Help-

Question Number 193116 by Mastermind last updated on 04/Jun/23 $$\mathrm{Show}\:\mathrm{that}\:\mathrm{for}\:\mathrm{all}\:\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\:\in\:\mathbb{R}\:\mathrm{with} \\ $$$$\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\:\geqslant\:\mathrm{0}\: \\ $$$$\left.\mathrm{1}\right)\:\sqrt{\mathrm{ab}}\sqrt{\mathrm{cd}}\:\leqslant\:\frac{\mathrm{1}}{\mathrm{4}}\left(\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} +\mathrm{d}^{\mathrm{2}} \right) \\ $$$$\left.\mathrm{2}\right)\:\left(\mathrm{abcd}\right)^{\frac{\mathrm{1}}{\mathrm{4}}} \:\leqslant\:\frac{\mathrm{1}}{\mathrm{4}}\left(\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}\right) \\ $$$$ \\ $$$$\mathrm{Help}!…

If-I-V-R-and-V-250-volts-and-R-50-ohms-Find-the-change-in-I-resulting-from-an-increase-of-1-volt-in-V-and-increase-of-0-5-ohm-in-R-

Question Number 130992 by benjo_mathlover last updated on 31/Jan/21 $$\mathrm{If}\:\mathrm{I}\:=\:\frac{\mathrm{V}}{\mathrm{R}}\:\mathrm{and}\:\mathrm{V}=\mathrm{250}\:\mathrm{volts}\:\mathrm{and}\:\mathrm{R}=\mathrm{50}\:\mathrm{ohms} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{change}\:\mathrm{in}\:\mathrm{I}\:\mathrm{resulting}\:\mathrm{from}\:\mathrm{an}\: \\ $$$$\mathrm{increase}\:\mathrm{of}\:\mathrm{1}\:\mathrm{volt}\:\mathrm{in}\:\mathrm{V}\:\mathrm{and}\:\mathrm{increase}\:\mathrm{of}\:\mathrm{0}.\mathrm{5}\:\mathrm{ohm} \\ $$$$\mathrm{in}\:\mathrm{R}. \\ $$ Answered by EDWIN88 last updated on 31/Jan/21…

Problem-Without-L-Hopital-calculate-lim-x-0-tan-2-x-x-2-cos-2x-x-2-sin-2-x-

Question Number 130925 by Chhing last updated on 30/Jan/21 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Problem} \\ $$$$\:\:\:\mathrm{Without}\:\:\mathrm{L}'\mathrm{Hopital} \\ $$$$\:\:\:\mathrm{calculate} \\ $$$$\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{tan}^{\mathrm{2}} \left(\mathrm{x}\right)−\mathrm{x}^{\mathrm{2}} \mathrm{cos}\left(\mathrm{2x}\right)}{\mathrm{x}^{\mathrm{2}} −\mathrm{sin}^{\mathrm{2}} \left(\mathrm{x}\right)} \\ $$…