Question Number 139587 by mathlove last updated on 29/Apr/21 Answered by qaz last updated on 29/Apr/21 $${f}\left(\mathrm{2}\right)=\int_{\mathrm{1}} ^{\mathrm{2}} {x}^{\mathrm{2}} +\mathrm{3}{xdx}+{f}\left(\mathrm{1}\right)=\left(\frac{\mathrm{1}}{\mathrm{3}}{x}^{\mathrm{3}} +\frac{\mathrm{3}}{\mathrm{2}}{x}^{\mathrm{2}} \right)\mid_{\mathrm{1}} ^{\mathrm{2}} +\mathrm{5}=\frac{\mathrm{71}}{\mathrm{6}} \\…
Question Number 139577 by bemath last updated on 29/Apr/21 $$\:\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{number}\frac{\left(\mathrm{x}−\mathrm{1}\right)^{\mathrm{2}} }{\mathrm{2}}+\frac{\left(\mathrm{y}−\mathrm{2}\right)^{\mathrm{2}} }{\mathrm{4}}+\frac{\left(\mathrm{z}−\mathrm{3}\right)^{\mathrm{2}} }{\mathrm{6}}+\mathrm{3}=\mid\mathrm{x}−\mathrm{1}\mid+\mid\mathrm{y}−\mathrm{2}\mid+\mid\mathrm{z}−\mathrm{3}\mid\: \\ $$ Answered by mr W last updated on 29/Apr/21 $${a}=\mid{x}−\mathrm{1}\mid\geqslant\mathrm{0},\:{b}=\mid{y}−\mathrm{2}\mid\geqslant\mathrm{0},\:{c}=\mid{z}−\mathrm{3}\mid\geqslant\mathrm{0} \\…
Question Number 74024 by ajfour last updated on 18/Nov/19 $$\begin{cases}{{h}^{\mathrm{2}} +{y}^{\mathrm{2}} +\left({k}−{z}\right)^{\mathrm{2}} ={s}^{\mathrm{2}} }\\{{a}^{\mathrm{2}} +\left({b}−{y}\right)^{\mathrm{2}} +{z}^{\mathrm{2}} ={s}^{\mathrm{2}} }\\{{ah}+{y}\left({y}−{b}\right)+{z}\left({z}−{k}\right)=\mathrm{0}}\\{\frac{{h}+{a}}{\mathrm{2}}+{yz}−\left({b}−{y}\right)\left({k}−{z}\right)=\mathrm{1}}\\{{b}+{a}\left({k}−{z}\right)+{hz}=\mathrm{1}}\\{{k}+{h}\left({b}−{y}\right)+{ay}=\mathrm{1}}\end{cases} \\ $$$${Find}\:\:{s}_{{min}} \:{or}\:{at}\:{least}\:{express} \\ $$$$\:{s}={f}\left({y}\right)\:{or}\:{g}\left({z}\right). \\ $$…
Question Number 139544 by melanie last updated on 28/Apr/21 Answered by bemath last updated on 28/Apr/21 $$\:\mathrm{Tanzalin}\:\mathrm{formula} \\ $$$$\:\begin{array}{|c|c|c|c|}{\mathrm{u}\left(\mathrm{diff}\right)}&\hline{\mathrm{dv}\:\left(\mathrm{integrate}\right)}\\{\mathrm{4x}}&\hline{\mathrm{cos}\:\left(\mathrm{2}−\mathrm{3x}\right)}\\{\mathrm{4}}&\hline{−\frac{\mathrm{1}}{\mathrm{3}}\mathrm{sin}\:\left(\mathrm{2}−\mathrm{3x}\right)}\\{\mathrm{0}}&\hline{−\frac{\mathrm{1}}{\mathrm{9}}\mathrm{cos}\:\left(\mathrm{2}−\mathrm{3x}\right)}\\\hline\end{array} \\ $$$$\mathrm{I}=\:−\frac{\mathrm{4x}\:\mathrm{sin}\:\left(\mathrm{2}−\mathrm{3x}\right)}{\mathrm{3}}\:+\frac{\mathrm{4}\:\mathrm{cos}\:\left(\mathrm{2}−\mathrm{3x}\right)}{\mathrm{9}}\:+\:\mathrm{c}\: \\ $$ Answered by…
Question Number 74006 by necxxx last updated on 17/Nov/19 $${If}\:\mathrm{3}{x}^{\mathrm{2}} {e}^{\mathrm{log}\:_{{x}} \mathrm{27}} =\mathrm{27000}\:{then}\:{find}\:{x} \\ $$ Commented by necxxx last updated on 17/Nov/19 $${please}\:{how}\:{can}\:{this}\:{be}\:{solved}\:{analytically} \\ $$$${or}\:{even}\:{by}\:{numerical}\:{methods}=…
Question Number 8468 by FilupSmith last updated on 12/Oct/16 $$\mathrm{Prove}\:\mathrm{or}\:\mathrm{disprove}\:\mathrm{that}: \\ $$$$\left(\mathrm{2}{k}+\mathrm{1}\right)^{{n}} \in\mathbb{O}\:\:\:\:\:\:\forall{k},{n}\in\mathbb{Z} \\ $$ Answered by Rasheed Soomro last updated on 12/Oct/16 $$\left(\mathrm{2k}+\mathrm{1}\right)^{\mathrm{n}} =\begin{pmatrix}{\mathrm{n}}\\{\mathrm{0}}\end{pmatrix}\left(\mathrm{2k}\right)^{\mathrm{n}}…
Question Number 8466 by FilupSmith last updated on 12/Oct/16 $${t}={n}^{\mathrm{3}} −{n}^{\mathrm{2}} \\ $$$${t}\in\mathbb{E} \\ $$$$\mathrm{Give}\:\mathrm{a}\:\mathrm{general}\:\mathrm{form}\:\mathrm{solving}\:\mathrm{for}\:{n} \\ $$$${i}.{e} \\ $$$${n}^{\mathrm{3}} −{n}^{\mathrm{2}} =\mathrm{2}{k} \\ $$$${n}=? \\ $$…
Question Number 139532 by mathdanisur last updated on 28/Apr/21 $$\frac{\mathrm{1}}{\mathrm{3}}\:−\:\frac{{x}−\mathrm{3}}{\mathrm{9}}\:+\:\frac{\left({x}−\mathrm{3}\right)^{\mathrm{2}} }{\mathrm{27}}\:+\:…\:=? \\ $$ Commented by mr W last updated on 28/Apr/21 $$=\frac{\mathrm{1}}{{x}} \\ $$ Commented…
Question Number 139505 by bemath last updated on 28/Apr/21 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}^{\mathrm{2}} \mathrm{y}^{\mathrm{3}} \: \\ $$$$\mathrm{when}\:\mathrm{3x}+\mathrm{4y}=\mathrm{5} \\ $$ Commented by mr W last updated on 28/Apr/21 $$\mathrm{3}{x}+\mathrm{4}{y}=\mathrm{5}\:\Rightarrow{y}\in\left(−\infty,\infty\right)…
Question Number 139502 by bemath last updated on 28/Apr/21 $$\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{8x}}\:\leqslant\:\mathrm{24}−\left(\mathrm{x}+\mathrm{2}\right)\left(\mathrm{x}+\mathrm{6}\right) \\ $$$$ \\ $$ Answered by TheSupreme last updated on 28/Apr/21 $${domain}:\:{x}<−\mathrm{8}\:\vee\:{x}>\mathrm{0} \\ $$$$\begin{cases}{\mathrm{24}−\left({x}+\mathrm{2}\right)\left({x}+\mathrm{6}\right)>\mathrm{0}\rightarrow\mathrm{12}−{x}^{\mathrm{2}}…