Question Number 101243 by Dwaipayan Shikari last updated on 01/Jul/20
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{solution}\: \\ $$$$\:\:\mathrm{xa}^{\frac{\mathrm{1}}{\mathrm{x}}} +\frac{\mathrm{1}}{\mathrm{x}}\mathrm{a}^{\mathrm{x}} =\mathrm{2a}\:\:\:\mathrm{a}\in\left\{−\mathrm{1},\mathrm{0},\mathrm{1}\right\}\:\:\:{and}\:{also}\:{find}\:{when}\:{a}\:\:{is}\:{not}\:{given} \\ $$
Commented by mr W last updated on 01/Jul/20
$${for}\:{a}>\mathrm{0}: \\ $$$${x}=\mathrm{1}\:{is}\:{the}\:{only}\:{solution}. \\ $$
Commented by mr W last updated on 01/Jul/20
$${a}=−\mathrm{1}: \\ $$$${x}\left(−\mathrm{1}\right)^{\frac{\mathrm{1}}{{x}}} +\frac{\left(−\mathrm{1}\right)^{{x}} }{{x}}=−\mathrm{2} \\ $$$$\Rightarrow{x}=\mathrm{1} \\ $$$$ \\ $$$${a}=\mathrm{0}: \\ $$$$\Rightarrow{x}\in{R}−\left\{\mathrm{0}\right\} \\ $$$$ \\ $$$${a}=\mathrm{1}: \\ $$$${x}+\frac{\mathrm{1}}{{x}}=\mathrm{2} \\ $$$$\Rightarrow{x}=\mathrm{1} \\ $$
Commented by Dwaipayan Shikari last updated on 01/Jul/20
$${You}\:{are}\:{right}\:{sir}.\:{Can}\:{you}\:{find}\:{when}\:{a}\:{is}\:{not}\:{given} \\ $$