Question Number 99117 by M±th+et+s last updated on 18/Jun/20
$${let}\:{a},{b},{c}\:\in\mathbb{R}\:{determine}\:{the}\:{minimum} \\ $$$${value} \\ $$$$ \\ $$$$\frac{\mathrm{3}{a}}{{b}+{c}}+\frac{\mathrm{4}{b}}{{a}+{c}}+\frac{\mathrm{5}{c}}{{a}+{b}} \\ $$
Answered by MJS last updated on 19/Jun/20
$$−\infty<\frac{\mathrm{3}{a}}{{b}+{c}}+\frac{\mathrm{4}{b}}{{a}+{c}}+\frac{\mathrm{5}{c}}{{a}+{b}}<+\infty \\ $$$$\mathrm{i}.\mathrm{e}.\:\mathrm{let}\:{a}=\mathrm{0}\wedge{b}=\mathrm{1} \\ $$$$\frac{\mathrm{3}{a}}{{b}+{c}}+\frac{\mathrm{4}{b}}{{a}+{c}}+\frac{\mathrm{5}{c}}{{a}+{b}}=\frac{\mathrm{5}{c}^{\mathrm{2}} +\mathrm{4}}{{c}} \\ $$$$\underset{{c}\rightarrow\pm\infty} {\mathrm{lim}}\frac{\mathrm{5}{c}^{\mathrm{2}} +\mathrm{4}}{{c}}\:=\pm\infty \\ $$