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Given-0-lt-a-lt-b-prove-that-b-a-2-8b-a-b-2-ab-b-a-2-8a-

Question Number 136739 by Ar Brandon last updated on 25/Mar/21 $$\mathrm{Given}\:\mathrm{0}<\mathrm{a}<\mathrm{b},\:\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{\left(\mathrm{b}−\mathrm{a}\right)^{\mathrm{2}} }{\mathrm{8b}}\leqslant\frac{\mathrm{a}+\mathrm{b}}{\mathrm{2}}−\sqrt{\mathrm{ab}}\leqslant\frac{\left(\mathrm{b}−\mathrm{a}\right)^{\mathrm{2}} }{\mathrm{8a}} \\ $$ Answered by snipers237 last updated on 26/Mar/21 $$\frac{{a}+{b}}{\mathrm{2}}−\sqrt{{ab}\:}=\:\frac{\left(\sqrt{{a}}−\sqrt{{b}}\right)^{\mathrm{2}}…

n-0-2n-n-1-4-n-2n-1-3-pi-3-48-pi-4-ln-2-2-

Question Number 136733 by Ñï= last updated on 25/Mar/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\begin{pmatrix}{\mathrm{2}{n}}\\{{n}}\end{pmatrix}\frac{\mathrm{1}}{\mathrm{4}^{{n}} \left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{3}} }=\frac{\pi^{\mathrm{3}} }{\mathrm{48}}+\frac{\pi}{\mathrm{4}}{ln}^{\mathrm{2}} \mathrm{2} \\ $$ Answered by mindispower last updated on 25/Mar/21…

A-particle-P-is-projected-from-a-point-O-at-the-edge-of-a-cliff-60m-from-the-sea-with-a-velocity-of-30ms-1-When-P-is-at-a-point-B-where-OB-is-a-horizontal-another-particle-Qsuch-that-P-and-Q-

Question Number 71198 by Rio Michael last updated on 13/Oct/19 $${A}\:{particle}\:{P}\:{is}\:{projected}\:{from}\:\:{a}\:{point}\:{O}\:{at}\:\:{the}\:{edge}\:{of}\:{a}\:{cliff}\:\mathrm{60}{m} \\ $$$${from}\:{the}\:{sea}\:{with}\:{a}\:{velocity}\:{of}\:\mathrm{30}{ms}^{−\mathrm{1}} .\:{When}\:{P}\:{is}\:{at}\:{a}\:{point}\:{B} \\ $$$${where}\:{OB}\:{is}\:{a}\:{horizontal},\:{another}\:{particle}\:{Qsuch}\:{that}\: \\ $$$${P}\:{and}\:{Q}\:{hit}\:{the}\:{sea}\:{simultaneously}\:{at}\:{thesame}\:{point}\:{A}.\:{Gven}\:{that}\:{they} \\ $$$${strike}\:{the}\:{sea}\:\mathrm{6}{seconds}\:{after}\:{P}\:{was}\:{fired}\bar {\:}\:{calculate} \\ $$$$\left.{a}\right)\:{the}\:{sine}\:{of}\:{the}\:{angle}\:{of}\:{elevation}\:{of}\:{projection}. \\ $$$$\left.{b}\right)\:{the}\:{distance}\:{from}\:{A}\:{to}\:{O}.…

the-curve-y-f-x-when-f-x-is-a-quadratic-expression-has-a-maximum-value-point-at-1-4-The-curve-touches-the-line-6x-y-13-Find-the-value-of-x-for-which-y-8-

Question Number 71196 by Rio Michael last updated on 12/Oct/19 $${the}\:{curve}\:{y}\:=\:{f}\left({x}\right),\:{when}\:{f}\left({x}\right)\:{is}\:{a}\:{quadratic}\:{expression}\:{has}\: \\ $$$${a}\:{maximum}\:{value}\:{point}\:{at}\:\left(\mathrm{1},\mathrm{4}\right).\:{The}\:{curve}\:{touches}\:{the}\:{line} \\ $$$$\mathrm{6}{x}\:+\:{y}\:=\:\mathrm{13}.\:{Find}\:{the}\:{value}\:{of}\:{x}\:{for}\:{which}\:{y}\:=\:\mathrm{8} \\ $$ Answered by MJS last updated on 12/Oct/19 $${f}\left({x}\right)\:\mathrm{is}\:\mathrm{quadratic}\:\Leftrightarrow\:{y}={ax}^{\mathrm{2}}…

Question-136734

Question Number 136734 by mohammad17 last updated on 25/Mar/21 Answered by Ñï= last updated on 25/Mar/21 $${Q}\mathrm{5}::: \\ $$$${y}_{{p}} =\frac{\mathrm{1}}{{D}^{\mathrm{2}} +\mathrm{3}{D}+\mathrm{12}}\left({e}^{{x}} \mathrm{cos}\:{x}−\mathrm{cos}\:{x}\right) \\ $$$$={e}^{{x}} \frac{\mathrm{1}}{{D}^{\mathrm{2}}…

calculus-I-prove-that-f-x-1-1-4x-n-0-2n-n-x-n-

Question Number 136723 by mnjuly1970 last updated on 25/Mar/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:….{calculus}\:\:\:\:\:\left({I}\right)….. \\ $$$$\:\:\:\:\:{prove}\:\:\:{that}\::: \\ $$$$\:\:\:\:{f}\left({x}\right)=\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}−\mathrm{4}{x}}}\:\overset{???} {=}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\begin{pmatrix}{\:\mathrm{2}{n}}\\{\:\:\:{n}}\end{pmatrix}\:{x}^{{n}} \\ $$$$ \\ $$ Answered by Dwaipayan Shikari…