Category: Relation and Functions

Question Number 209707 by alcohol last updated on 19/Jul/24 $$a,b\in \mathbb{C}:a\stackrel{-}{b}+b=0f:z^{\prime }=a\stackrel{-}{z}+b$$$$suchthatf\left(M\right)=M^{\prime }$$$$1.let{z}_A=zand{z}_{A^{\prime }}=z^{\prime }andf\left(A\right)=A$$$$showthat2Re\left(\stackrel{-}{bz}\right)=b\stackrel{-}{b}$$$$\left({A}\:{is}\:{the}\:{set}\:{of}\:{invariant}\:{points}\:{and}\right. \…

Question Number 208431 by alcohol last updated on 15/Jun/24 $$h_a\left(x\right)=e^{-x}+{ax}^2$$$$showthat{h}_{a}admitsaminimumin\mathbb{R}$$ Answered by mathzup last updated on 15/Jun/24 $${cas}\:\mathrm{1}\:\:\:\:\:\:{a}>\mathrm{0}…

Question Number 208103 by depressiveshrek last updated on 05/Jun/24 $$\mathrm{Find}\inf \{\frac{m}{n}\mid m,n\in \mathbb{N},m<n-2\}$$ Answered by A5T last updated on 05/Jun/24 $$Sincem,n\in \mathbb{N},\frac{m}{n}>0,so0isalowerbound.$$$$\frac{m}{n}⩾\frac{1}{n}and\frac{1}{n}isintheset.$$$${For}\:{any}\:\epsilon>\mathrm{0},\:{one}\:{can}\:{always}\:{find}\:{n}\:{such}\:{that} \…

Question Number 207314 by universe last updated on 11/May/24 $$\mathrm{let}\mathrm{f}:\mathbb{R}\to \mathbb{R}\mathrm{be}\mathrm{a}\mathrm{continuous}\mathrm{function}\mathrm{then}$$$$\mathrm{show}\mathrm{that}$$$$\left(1\right)\mathrm{if}\mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}\left({\mathrm{x}}^2\right)\mathrm{\forall }\mathrm{x}\in \mathbb{R}\mathrm{then}\mathrm{f}\mathrm{is}\mathrm{a}\mathrm{constant}$$$$\mathrm{function}$$$$\left(2\right)\mathrm{if}\mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}(2\mathrm{x}+1)\mathrm{\forall }\mathrm{x}\in \mathbb{R}\mathrm{then}\mathrm{f}\mathrm{is}\mathrm{a}$$$$\mathrm{constant}\mathrm{function}$$ Answered by…

Question Number 206993 by necx122 last updated on 03/May/24 $$Ifthenthtermofasequenceis$$$$givenby\frac{n^2-2n}{4},whatisthesumofn$$$$termsofthesequence?$$ Answered by Rasheed.Sindhi last updated on 03/May/24 $$\underset{{k}=\mathrm{1}}…