Question Number 72830 by Tip Top last updated on 03/Nov/19
$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:{x}\:\mathrm{satisfying}\:\mathrm{the}\: \\ $$$$\mathrm{inequalities}\:\frac{\left(\mathrm{2}{x}−\mathrm{1}\right)\left({x}−\mathrm{1}\right)^{\mathrm{4}} \left({x}−\mathrm{2}\right)^{\mathrm{4}} }{\left({x}−\mathrm{2}\right)\left({x}−\mathrm{4}\right)^{\mathrm{4}} }\leqslant\:\mathrm{0} \\ $$
Answered by MJS last updated on 03/Nov/19
$$\mathrm{defined}\:\mathrm{for}\:\mathbb{R}\backslash\left\{\mathrm{2},\:\mathrm{4}\right\} \\ $$$$\frac{\left({x}−\mathrm{1}\right)^{\mathrm{4}} \left({x}−\mathrm{2}\right)^{\mathrm{4}} }{\left({x}−\mathrm{4}\right)^{\mathrm{4}} }\geqslant\mathrm{0}\:\forall\:{x}\in\mathbb{R}\backslash\left\{\mathrm{4}\right\} \\ $$$$\frac{\mathrm{2}{x}−\mathrm{1}}{{x}−\mathrm{2}}\leqslant\mathrm{0} \\ $$$$\mathrm{2}{x}−\mathrm{1}\geqslant\mathrm{0}\wedge{x}−\mathrm{2}<\mathrm{0}\:\vee\:\mathrm{2}{x}−\mathrm{1}\leqslant\mathrm{0}\wedge{x}−\mathrm{2}>\mathrm{0} \\ $$$$\Rightarrow\:\frac{\mathrm{1}}{\mathrm{2}}\leqslant{x}<\mathrm{2} \\ $$
Answered by Tanmay chaudhury last updated on 03/Nov/19
$$\left({x}−\mathrm{1}\right)^{\mathrm{4}} ,\left({x}−\mathrm{2}\right)^{\mathrm{4}} {and}\:\left({x}−\mathrm{4}\right)^{\mathrm{4}} \:\:{even}\:{power} \\ $$$${so}\:{they}\:{are}\:{always}\:+{ve}\:. \\ $$$${x}\neq\mathrm{2}\:\:\:\:\:{x}\neq\:\mathrm{4} \\ $$$${critical}\:{value}\:{of}\:{x}=\frac{\mathrm{1}}{\mathrm{2}}=\mathrm{0}.\mathrm{5}\:{and}\:\mathrm{2} \\ $$$${f}\left({x}\right)=\frac{\mathrm{2}{x}−\mathrm{1}}{\left({x}−\mathrm{2}\right)}×\left[\frac{\left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)}{\left({x}−\mathrm{4}\right)}\right]^{\mathrm{4}} \\ $$$${when}\:{x}>\mathrm{2}\:\:{f}\left({x}\right)>\mathrm{0} \\ $$$${when}\:{x}<\mathrm{0}.\mathrm{5}\:\:{f}\left({x}\right)>\mathrm{0} \\ $$$${when}\:\:\:\:\:\mathrm{2}>{x}>\mathrm{0}.\mathrm{5}\:{f}\left({x}\right)<\mathrm{0} \\ $$$${solution}\:\:\:{x}\in\:\left[\mathrm{0}.\mathrm{5},\:\mathrm{2}\right)\:\:\boldsymbol{{pls}}\:\boldsymbol{{check}} \\ $$$$ \\ $$