Question Number 4702 by 314159 last updated on 22/Feb/16 $$\mathrm{Let}\:\mathrm{f}\:\mathrm{and}\:\mathrm{g}\:\mathrm{be}\:\mathrm{functions}\:\mathrm{such}\:\mathrm{that}\:\mathrm{for}\:\mathrm{all} \\ $$$$\mathrm{real}\:\mathrm{number}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y},\mathrm{g}\left(\mathrm{f}\left(\mathrm{x}+\mathrm{y}\right)\right)=\mathrm{f}\left(\mathrm{x}\right)+\left(\mathrm{x}+\mathrm{y}\right)\mathrm{g}\left(\mathrm{y}\right). \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{g}\left(\mathrm{0}\right)+\mathrm{g}\left(\mathrm{1}\right)+\mathrm{g}\left(\mathrm{2}\right)+\mathrm{g}\left(\mathrm{3}\right)+…+\mathrm{g}\left(\mathrm{2016}\right). \\ $$ Commented by prakash jain last updated on 22/Feb/16 $$\mathrm{Let}\:\mathrm{us}\:\mathrm{try}\:\mathrm{for}\:\mathrm{trivial}\:\mathrm{solution}…
Question Number 4690 by thachan last updated on 21/Feb/16 $${x}+{y}=\mathrm{67} \\ $$$${x}−{ay}=\mathrm{7} \\ $$ Answered by 123456 last updated on 21/Feb/16 $$−{x}+{ay}=−\mathrm{7} \\ $$$$\left({a}+\mathrm{1}\right){y}=\mathrm{60} \\…
Question Number 70219 by Mr. K last updated on 02/Oct/19 Commented by MJS last updated on 02/Oct/19 $${a}=\mathrm{54}° \\ $$$${b}=\mathrm{101}° \\ $$$${c}=\mathrm{126}° \\ $$$${d}=\mathrm{79}° \\…
Question Number 4683 by thachan last updated on 21/Feb/16 $$\int_{\boldsymbol{{sinx}}} ^{\mathrm{sin}\:\boldsymbol{{x}}+\mathrm{2}} \left(\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{ax}}+\mathrm{cos}\:\left(\mathrm{6}\boldsymbol{{x}}\right)\right)\boldsymbol{{dx}} \\ $$ Commented by prakash jain last updated on 21/Feb/16 $$\mathrm{The}\:\mathrm{bounds}\:\mathrm{are}\:\mathrm{dependent}\:\mathrm{on}\:{x}? \\…
Question Number 135744 by liberty last updated on 15/Mar/21 $$ \\ $$Shabbat Shalom~ As Passover approaches What are you in bondage from? Don’t blame others,…
Question Number 4679 by 314159 last updated on 22/Feb/16 $${Let}\:{a},{b},{c},{d}\:\in{R}\:,{a}+{b}+{c}+{d}=\mathrm{0}.{Prove}\:{that} \\ $$$$\underset{{cyc}} {\sum}\left(\frac{\mathrm{1}}{\:\sqrt{{a}^{\mathrm{2}} +{a}+\mathrm{1}}}\right)\leqslant\mathrm{4}. \\ $$ Commented by prakash jain last updated on 21/Feb/16 $$\mathrm{It}\:\mathrm{is}\:\mathrm{not}\:\mathrm{much}\:\mathrm{less}\:\mathrm{than}\:\mathrm{4}.…
Question Number 4671 by 314159 last updated on 20/Feb/16 $${Define}\:{a}\:{function}\:{f}:{R}\rightarrow{R}\:{such}\:{that}\:{f}\left({f}\left({x}\right)\right)={x}^{\mathrm{2}} −{x}+\mathrm{1} \\ $$$${for}\:{all}\:{real}\:{x}.{Evaluate}\:{f}\left(\mathrm{0}\right). \\ $$$$ \\ $$ Answered by 123456 last updated on 20/Feb/16 $${f}\left({f}\left({x}\right)\right)={x}^{\mathrm{2}}…
Question Number 70196 by smartsmith459@gmail.com last updated on 01/Oct/19 Answered by MJS last updated on 01/Oct/19 $${x}={a}^{\mathrm{log}_{{b}} \:{c}} ={a}^{\frac{\mathrm{ln}\:{c}}{\mathrm{ln}\:{b}}} \\ $$$$\mathrm{ln}\:{x}\:=\frac{\mathrm{ln}\:{c}}{\mathrm{ln}\:{b}}\mathrm{ln}\:{a}\:\Rightarrow\:{x}={a}^{\mathrm{log}_{{b}} \:{c}} ={c}^{\mathrm{log}_{{b}} \:{a}} \\…
Question Number 70198 by smartsmith459@gmail.com last updated on 01/Oct/19 Answered by MJS last updated on 01/Oct/19 $${x}={a}^{\sqrt{\mathrm{log}_{{a}} \:{b}}} \\ $$$$\mathrm{ln}\:{x}\:=\sqrt{\frac{\mathrm{ln}\:{b}}{\mathrm{ln}\:{a}}}\mathrm{ln}\:{a}\:=\sqrt{\mathrm{ln}\:{a}\:×\mathrm{ln}\:{b}} \\ $$$$\Rightarrow\:\mathrm{5}^{\sqrt{\mathrm{log}_{\mathrm{5}} \:\mathrm{7}}} −\mathrm{7}^{\sqrt{\mathrm{log}_{\mathrm{7}} \:\mathrm{5}}}…
Question Number 70197 by smartsmith459@gmail.com last updated on 01/Oct/19 Commented by mr W last updated on 01/Oct/19 $${in}\:{the}\:{first}\:\mathrm{27}\:{days}\:{it}\:{climbs}\:\mathrm{27}{m}. \\ $$$${in}\:{the}\:\mathrm{28}{th}\:{day}\:{it}\:{escapes}\:{from}\:{the} \\ $$$${pit}.\:\Rightarrow{totally}\:\mathrm{28}\:{days} \\ $$ Terms…