Question Number 134432 by liberty last updated on 03/Mar/21 $$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{and}\:\mathrm{minimum} \\ $$$$\mathrm{value}\:\mathrm{of}\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} \:\mathrm{subject}\:\mathrm{to}\: \\ $$$$\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{4}}+\frac{\mathrm{y}^{\mathrm{2}} }{\mathrm{5}}+\frac{\mathrm{z}^{\mathrm{2}} }{\mathrm{25}}\:=\:\mathrm{1}\:\mathrm{and}\:\mathrm{z}\:=\:\mathrm{x}+\mathrm{y}\:. \\ $$ Terms of Service…
Question Number 134435 by mnjuly1970 last updated on 03/Mar/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:….\mathscr{A}{dvanced}\:\:{calculus}…. \\ $$$$\:\:\:\:{prove}\:{that}:: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{P}=\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\left(\mathrm{1}−\frac{\mathrm{1}}{{n}^{\mathrm{4}} }\right)=\frac{{sin}\left(\pi\right).{sinh}\left(\pi\right)}{\pi^{\:\mathrm{2}} }\:..\checkmark \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:………….\:……….. \\ $$ Answered by Dwaipayan…
Question Number 3330 by Filup last updated on 11/Dec/15 $$\mathrm{If}\:\mathrm{you}\:\mathrm{had}\:\mathrm{a}\:\mathrm{rectangle}\:\mathrm{with}\:\mathrm{side}\:\mathrm{lengths} \\ $$$${h}\:\mathrm{and}\:{b},\:\mathrm{the}\:\mathrm{area}\:\mathrm{is}\:\mathrm{given}\:\mathrm{by}: \\ $$$${A}\left({h},\:{b}\right)={hb} \\ $$$$ \\ $$$$\mathrm{If}\:\mathrm{I}\:\mathrm{were}\:\mathrm{to}\:\mathrm{change}\:\left(\mathrm{increase}\:\mathrm{or}\:\mathrm{decrease}\right) \\ $$$$\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{one}\:\mathrm{or}\:\mathrm{both}\:\mathrm{of}\:\mathrm{the}\:\mathrm{variables} \\ $$$$\mathrm{in}\:{A},\:\mathrm{how}\:\mathrm{can}\:\mathrm{I}\:\mathrm{write}\:\mathrm{the}\:\mathrm{change}\:\mathrm{in}\:\mathrm{area}? \\ $$ Answered…
Question Number 3324 by prakash jain last updated on 10/Dec/15 $$\mathrm{Prove}\:\mathrm{that}\:\mathrm{among}\:\mathrm{all}\:\mathrm{triangle}\:\mathrm{of}\:\mathrm{equal} \\ $$$$\mathrm{perimeter},\:\mathrm{equilateral}\:\mathrm{triangle}\:\mathrm{has}\:\mathrm{the} \\ $$$$\mathrm{largest}\:\mathrm{area}. \\ $$ Commented by 123456 last updated on 10/Dec/15 $$\mathrm{2}{s}\left({x},{y},{z}\right)={x}+{y}+{z}…
Question Number 134391 by bramlexs22 last updated on 03/Mar/21 $$\mathrm{Given}\:\mathrm{f}\left(\mathrm{x}\right)=\int_{\mathrm{0}} ^{\:{x}} \:\left(\frac{{t}^{\mathrm{4}} −{t}^{\mathrm{2}} }{{t}^{\mathrm{2}} +\mathrm{1}}\right)\:{dt}. \\ $$$$\mathrm{Find}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{f}\left(\mathrm{x}\right). \\ $$ Answered by liberty last updated on…
Question Number 134373 by mnjuly1970 last updated on 02/Mar/21 Answered by Dwaipayan Shikari last updated on 02/Mar/21 $${I}\left({a}\right)=\underset{{n}=−\infty} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left({n}^{\mathrm{2}} +{a}^{\mathrm{2}} \right)^{\mathrm{2}} }=\frac{\mathrm{1}}{{a}^{\mathrm{4}} }+\mathrm{2}\underset{{n}=\mathrm{1}} {\overset{\infty}…
Question Number 3228 by Rasheed Soomro last updated on 08/Dec/15 $$\mathcal{W}{hy}\:{is}\:{circumference}\:{of}\:{a}\:{circle}\:{derivative}\:{of} \\ $$$${its}\:{area}? \\ $$$$\frac{{d}}{{dr}}\left(\pi{r}^{\mathrm{2}} \right)=\mathrm{2}\pi{r}={circumference}. \\ $$ Commented by 123456 last updated on 08/Dec/15…
Question Number 134285 by bramlexs22 last updated on 02/Mar/21 $$\mathrm{Find}\:\mathrm{maximum}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\mathrm{f}\left(\mathrm{x}\right)\:=\:\sqrt{\left(\mathrm{16}−\mathrm{x}^{\mathrm{2}} \right)\left(\mathrm{x}^{\mathrm{2}} −\mathrm{9}\right)}\: \\ $$ Answered by EDWIN88 last updated on 02/Mar/21 $$\mathrm{let}\:\mathrm{g}\left(\mathrm{x}\right)=\left(\mathrm{16}−\mathrm{x}^{\mathrm{2}} \right)\left(\mathrm{x}^{\mathrm{2}}…
Question Number 68703 by Maclaurin Stickker last updated on 15/Sep/19 $$\frac{{d}}{{dx}}\left({ln}\left(\sqrt{\frac{{x}^{\mathrm{2}} −\mathrm{1}}{{x}^{\mathrm{2}} +\mathrm{1}}}\right)\right)=? \\ $$ Commented by MJS last updated on 15/Sep/19 $$\frac{{d}}{{dx}}\left[\mathrm{ln}\:\sqrt{\frac{{x}^{\mathrm{2}} −\mathrm{1}}{{x}^{\mathrm{2}} +\mathrm{1}}}\right]=\frac{\mathrm{1}}{\mathrm{2}}×\frac{{d}}{{dx}}\left[\mathrm{ln}\:\frac{{x}^{\mathrm{2}}…
Question Number 134204 by liberty last updated on 01/Mar/21 $$\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\sqrt{\mathrm{2}\left(\mathrm{1}−\mathrm{cos}\:\mathrm{2x}\right)}\:. \\ $$$$\mathrm{find}\:\mathrm{f}\:'\left(\mathrm{0}\right). \\ $$ Answered by bramlexs22 last updated on 02/Mar/21 $$\:\mathrm{f}\left(\mathrm{x}\right)=\sqrt{\mathrm{2}\left(\mathrm{2sin}\:^{\mathrm{2}} \mathrm{x}\right)}\:=\mid\mathrm{4sin}\:\mathrm{x}\:\mid=\mathrm{2}\mid\mathrm{sin}\:\mathrm{x}\:\mid \\ $$$$\:\mathrm{f}\:'\left(\mathrm{0}^{−}…