Category: Algebra

Question Number 156107 by cortano last updated on 08/Oct/21 $${(1+\frac{1}{\mathrm{x}})}^{\mathrm{x}+1}={(1+\frac{1}{2019})}^{2019}$$ Commented by mr W last updated on 08/Oct/21 $${(1+\frac{1}{x})}^{x+1}$$$$=\left(\frac{{x}+\mathrm{1}}{{x}}\right)^{{x}+\mathrm{1}} \…

Question Number 25025 by tawa tawa last updated on 01/Dec/17 $$\mathrm{If}{\mathrm{a}}^4+{\mathrm{b}}^4+{\mathrm{c}}^4+{\mathrm{d}}^4=16,\mathrm{prove}\mathrm{that}:{\mathrm{a}}^5+{\mathrm{b}}^5+{\mathrm{c}}^5+{\mathrm{d}}^5⩽32$$$$\mathrm{for}\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\in \mathbb{R}$$ Answered by…

Question Number 25023 by Tinkutara last updated on 01/Dec/17 $$\mathrm{Consider}\mathrm{the}\mathrm{function}f\left(x\right)\mathrm{which}$$$$\mathrm{satisfying}\mathrm{the}\mathrm{functional}\mathrm{equation}$$$$2f\left(x\right)+f(1-x)=x^2+1,\mathrm{\forall }x\in R$$$$\mathrm{and}g\left(x\right)=3f\left(x\right)+1.\mathrm{The}\mathrm{range}\mathrm{of}$$$$\varphi \left(x\right)=g\left(x\right)+\frac{1}{g\left(x\right)+1}\mathrm{is}$$ Answered by prakash jain…

Question Number 156086 by MathSh last updated on 07/Oct/21 $${\psi }^{\left(1\right)}\left(\frac{1}{6}\right)-{\psi }^{\left(1\right)}\left(\frac{5}{6}\right)=10{\psi }^{\left(1\right)}\left(\frac{1}{3}\right)-\frac{20}{3}{\pi }^2$$$$$$ Commented by puissant last updated on 09/Oct/21 $$\psi^{\left(\mathrm{1}\right)}…

Question Number 25001 by Tinkutara last updated on 30/Nov/17 $$\mathrm{If}x,y>0,\mathrm{then}\mathrm{the}\mathrm{minimum}\mathrm{value}\mathrm{of}$$$$2x^2+\frac{2}{x}-2x+2y^2+\frac{2}{y}-2y+2\mathrm{is}$$$$\mathrm{equal}\mathrm{to}$$ Commented by prakash jain last updated on…