Question Number 131332 by ZiYangLee last updated on 03/Feb/21 $$\mathrm{If}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{function}\:\mathrm{defined}\:\mathrm{for}\:\mathrm{all}\:\mathrm{real}\:\mathrm{numbers} \\ $$$$\mathrm{such}\:\mathrm{that}\:{f}\left(\mathrm{2}{x}\right)={x}+{f}\left({x}\right)\:\mathrm{and}\:{f}\left(\mathrm{1}\right)=\mathrm{2}, \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{f}\left(\mathrm{64}\right). \\ $$$$\left[\mathrm{Ans}:\:\mathrm{65}\right] \\ $$ Answered by mr W last updated on…
Question Number 261 by raj last updated on 25/Jan/15 $$\mathrm{If}\:\underset{\mathrm{0}} {\overset{{x}} {\int}}{f}\left({t}\right){dt}={x}+\underset{{x}} {\overset{\mathrm{1}} {\int}}{tf}\left({t}\right){dt},\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value} \\ $$$$\mathrm{of}\:{f}\left(\mathrm{1}\right). \\ $$ Answered by prakash jain last updated on…
Question Number 131334 by Algoritm last updated on 03/Feb/21 Answered by bluberry508 last updated on 06/Feb/21 $$ \\ $$$${let}\:\:{g}\left({x}\right)=\int_{\mathrm{0}} ^{\:{x}} {xf}\:'\left({t}\right)−{tf}^{\:'} \left({t}\right){dt}\:= \\ $$$${x}\int_{\mathrm{0}} ^{\:{x}}…
Question Number 65797 by Souvik Ghosh last updated on 04/Aug/19 $${find}\:{the}\:{constant}\:\:{a},{b}\:{and}\:\:{c}\:\:{so} \\ $$$${that}\:{the}\:{direction}\:{derivative}\:{of} \\ $$$$\Phi={axy}^{\mathrm{2}} +{byz}+{cz}^{\mathrm{2}} {x}^{\mathrm{3}} \:{at}\:\left(\mathrm{1},\mathrm{2},−\mathrm{1}\right) \\ $$$${has}\:{a}\:{maximum}\:{of}\:{magnitude} \\ $$$$\mathrm{64}\:{jn}\:{a}\:{direction}\:{parallel}\:{to}\:{the} \\ $$$${z}\:{axis}. \\…
Question Number 260 by 9999 last updated on 25/Jan/15 $$\int_{−\mathrm{1}} ^{+\mathrm{1}} \mid\mathrm{1}−{x}\mid{dx}= \\ $$ Answered by 123456 last updated on 17/Dec/14 $$\mid\mathrm{1}−{x}\mid=\begin{cases}{\mathrm{1}−{x}}&{{x}\leqslant\mathrm{1}}\\{{x}−\mathrm{1}}&{{x}\geqslant\mathrm{1}}\end{cases} \\ $$$$\int_{−\mathrm{1}} ^{+\mathrm{1}}…
Question Number 259 by a@b.c last updated on 25/Jan/15 $$\int_{{a}} ^{{b}} \:\frac{{f}\left({x}\right)}{{f}\left({x}\right)+{f}\left({a}+{b}−{x}\right)}{dx}= \\ $$ Answered by prakash jain last updated on 17/Dec/14 $$\mathrm{Substitue}\:{x}={a}+{b}−{y}\:\Rightarrow{dx}=−{dy} \\ $$$$\mathrm{The}\:\mathrm{given}\:\mathrm{integral}\:{I}…
Question Number 131328 by rs4089 last updated on 03/Feb/21 Commented by MJS_new last updated on 04/Feb/21 $$\mathrm{really}? \\ $$$$\mathrm{just}\:\mathrm{try}\:\mathrm{some}\:\mathrm{values}\:\mathrm{with}\:{n}={k}^{\mathrm{2}} \wedge{k}\in\mathbb{N} \\ $$$${y}_{{k}} =\frac{\left(\mathrm{2}{k}^{\mathrm{2}} \right)!}{\mathrm{2}^{{k}} }…
Question Number 258 by mchawla last updated on 25/Jan/15 $$\mathrm{If}\:\mathrm{log}_{\mathrm{3}} \mathrm{2},\:\mathrm{log}_{\mathrm{3}} \left(\mathrm{2}^{{x}} −\mathrm{5}\right)\:\mathrm{and}\:\mathrm{log}_{\mathrm{3}} \left(\mathrm{2}^{{x}} −\frac{\mathrm{7}}{\mathrm{2}}\right)\:\mathrm{are} \\ $$$$\mathrm{in}\:\mathrm{arithmetic}\:\mathrm{progression}\:\mathrm{then}\:\mathrm{find} \\ $$$$\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{x}. \\ $$ Answered by 123456 last…
Question Number 257 by abcd last updated on 25/Jan/15 $$\mathrm{If}\:\mathrm{f}\left({x}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{function}\:\mathrm{satisfying} \\ $$$$\mathrm{f}\left(\mathrm{x}+\mathrm{y}\right)=\mathrm{f}\left(\mathrm{x}\right)\mathrm{f}\left(\mathrm{y}\right)\:\mathrm{for}\:\mathrm{all}\:\mathrm{x},\mathrm{y}\in\mathbb{N}\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{f}\left(\mathrm{1}\right)=\mathrm{3}\:\mathrm{and}\:\underset{{x}=\mathrm{1}} {\overset{{n}} {\sum}}\mathrm{f}\left(\mathrm{x}\right)=\mathrm{120}.\:\mathrm{Then}\:\mathrm{find} \\ $$$$\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{n}. \\ $$ Answered by 123456 last updated…
Question Number 131325 by mohammad17 last updated on 03/Feb/21 Commented by mohammad17 last updated on 03/Feb/21 $${find}\:{n} \\ $$ Answered by mr W last updated…