Question Number 76290 by benjo last updated on 26/Dec/19
$$\mathrm{how}\:\mathrm{to}\:\mathrm{prove}\:\mathrm{x}^{\mathrm{y}} \:+\mathrm{y}^{\mathrm{x}} \:\geqslant\mathrm{1}\:,\:\mathrm{x},\mathrm{y}\:\in\mathbb{R} \\ $$$$\mathrm{x},\mathrm{y}\:>\:\mathrm{0} \\ $$
Commented by MJS last updated on 26/Dec/19
$$\mathrm{we}\:\mathrm{can}\:\mathrm{exchange}\:{x}\:\mathrm{and}\:{y}\:\Rightarrow\:\mathrm{let}\:{x}\leqslant{y} \\ $$$$\mathrm{it}'\mathrm{s}\:\mathrm{easy}\:\mathrm{for}\:\mathrm{these}\:\mathrm{cases} \\ $$$$\left(\mathrm{1}\right)\:{x}={y} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{1}\leqslant{x}<{y}<\infty \\ $$$$\left(\mathrm{3}\right)\:\mathrm{0}<{x}\leqslant\mathrm{1}<{y}<\infty \\ $$$$\mathrm{I}\:\mathrm{found}\:\mathrm{no}\:\mathrm{proper}\:\mathrm{way}\:\mathrm{for} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{0}<{x}<{y}<\mathrm{1} \\ $$
Answered by mind is power last updated on 26/Dec/19
$$\mathrm{x}=\mathrm{y}=−\mathrm{3}\:\mathrm{false} \\ $$$$ \\ $$
Commented by benjo last updated on 26/Dec/19
$$\mathrm{sir}\:\mathrm{x}\:>\mathrm{0}\:\mathrm{and}\:\mathrm{y}>\mathrm{0} \\ $$