Question Number 34365 by mondodotto@gmail.com last updated on 05/May/18
$$\boldsymbol{\mathrm{if}}\:\boldsymbol{\mathrm{a}}>\boldsymbol{\mathrm{b}}>\mathrm{0}\:\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}} \\ $$$$\boldsymbol{\mathrm{b}}<\frac{\boldsymbol{\mathrm{ae}}^{\boldsymbol{{x}}} +\boldsymbol{\mathrm{be}}^{−\boldsymbol{{x}}} }{\boldsymbol{\mathrm{e}}^{\boldsymbol{{x}}} +\boldsymbol{\mathrm{e}}^{−\boldsymbol{{x}}} }<\boldsymbol{\mathrm{a}} \\ $$
Answered by MJS last updated on 05/May/18
$${b}\mathrm{e}^{{x}} +{b}\mathrm{e}^{−{x}} <{a}\mathrm{e}^{{x}} +{b}\mathrm{e}^{−{x}} \:\Rightarrow \\ $$$$\Rightarrow\:{b}\mathrm{e}^{{x}} <{a}\mathrm{e}^{{x}} \\ $$$${a}\mathrm{e}^{{x}} +{b}\mathrm{e}^{−{x}} <{a}\mathrm{e}^{{x}} +{a}\mathrm{e}^{−{x}} \:\Rightarrow \\ $$$$\Rightarrow\:{b}\mathrm{e}^{−{x}} <{a}\mathrm{e}^{−{x}} \\ $$$$\mathrm{both}\:\mathrm{are}\:\mathrm{true} \\ $$