Category: Algebra

Question Number 195288 by CrispyXYZ last updated on 29/Jul/23 $$x,y,z\in {\mathbb{R}}_{+},$$$$P=\frac{x}{x+y}+\frac{y}{y+z}+\frac{z}{z+x},$$$$Q=\frac{y}{x+y}+\frac{z}{y+z}+\frac{x}{z+x},$$$$Q=\frac{z}{x+y}+\frac{x}{y+z}+\frac{y}{z+x}.$$$$f(x,y,z)=\max \{P,Q,R\},\mathrm{find}{f}_{\min }.$$ Commented by Frix…

Question Number 195227 by York12 last updated on 28/Jul/23 $$$$$$\alpha_{\mathrm{1}} ^{\mathrm{3}} \left[\frac{\underset{{i}=\mathrm{2}} {\overset{{n}} {\prod}}\left({x}−\alpha_{{i}} \right)}{\underset{{i}=\mathrm{2}} {\overset{{n}} {\prod}}\left(\alpha_{\mathrm{1}} −\alpha_{{i}} \right)}\right]+\underset{{j}=\mathrm{2}} {\overset{{n}} {\sum}}\left(\alpha_{{j}} ^{\mathrm{3}} \left[\frac{\underset{{i}=\mathrm{1}}…

Question Number 195255 by Spillover last updated on 28/Jul/23 $$\int \frac{dx}{\cos {}^{3}x\sqrt{4\sin x\cos x}}$$ Answered by Frix last updated on 28/Jul/23 $$[\mathrm{Using}t=\sqrt{\tan x}]$$$$=\int\left({t}^{\mathrm{4}} +\mathrm{1}\right){dt}=…=\frac{\mathrm{5}+\mathrm{tan}^{\mathrm{2}} \:{x}}{\mathrm{5}}\sqrt{\mathrm{tan}\:{x}}\:+{C}…

Question Number 195254 by Spillover last updated on 02/Aug/23 $${\int }^{\mathit{s}\mathit{p}\mathit{i}\mathit{l}\mathit{l}\mathit{o}\mathit{v}\mathit{e}\mathit{r}}\frac{\sin {}^{2}x\cos {}^{2}x}{{(\sin {}^{5}x+\cos {}^{3}x\sin {}^{2}x+\sin {}^{3}x\cos {}^{2}x+\cos {}^{5}x)}^2}dx$$ Answered by Spillover…

Question Number 195251 by Spillover last updated on 28/Jul/23 $$If{x}^{[16{\left(\log {}_{5}x\right)}^3-68\log {}_{5}x]}=5^{-16}$$$$thenFindthetheproductofx$$$$$$ Commented by Frix last…