Question Number 32897 by artibunja last updated on 06/Apr/18 | ||
$$ \\ $$ $${e}^{\pi} \:>\:\pi^{{e}} \:\:{or}\:\:\:{e}^{\pi} <\pi^{{e}} \:? \\ $$ $$ \\ $$ | ||
Answered by MJS last updated on 06/Apr/18 | ||
$${e}^{{x}} ={x}^{{e}} ;\:{x}>\mathrm{1}\:\Rightarrow\:{x}={e}\:\Rightarrow\:\forall{x}>{e}:\:{e}^{{x}} >{x}^{{e}} \\ $$ $$\pi>{e}\:\Rightarrow\:{e}^{\pi} >\pi^{{e}} \\ $$ | ||
Commented byMJS last updated on 06/Apr/18 | ||
$$\mathrm{interestingly}\:\mathrm{for}\:{a}^{{x}} ={x}^{{a}} \:\mathrm{with}\:{a}>\mathrm{1} \\ $$ $$\mathrm{and}\:{x}>\mathrm{1}\:\mathrm{two}\:\mathrm{solutions}\:\mathrm{seem}\:\mathrm{to} \\ $$ $$\mathrm{exist}\:\left(\mathrm{i}.\mathrm{e}.\:\mathrm{2}^{{x}} ={x}^{\mathrm{2}} \:\Rightarrow\:{x}=\mathrm{2}\vee{x}=\mathrm{4};\right. \\ $$ $$\left.\mathrm{3}^{{x}} ={x}^{\mathrm{3}} \:\Rightarrow\:{x}=\mathrm{3}\vee{x}\approx\mathrm{2}.\mathrm{47805}\right)\:\mathrm{but} \\ $$ $$\mathrm{for}\:{a}={e}\:\mathrm{there}'\mathrm{s}\:\mathrm{only}\:\mathrm{one}:\:{x}={e} \\ $$ $$ \\ $$ $$\mathrm{can}\:\mathrm{anybody}\:\mathrm{further}\:\mathrm{explain}\:\mathrm{this}? \\ $$ | ||
Commented bymrW2 last updated on 06/Apr/18 | ||
$${solution}\:{for}\:{a}^{{x}} ={x}^{{a}} \:\left({a}>\mathrm{0}\right): \\ $$ $${a}^{\frac{{x}}{{a}}} ={x} \\ $$ $${e}^{\frac{{x}}{{a}}\:\mathrm{ln}\:{a}} ={x} \\ $$ $$\left(−\frac{{x}}{{a}}\:\mathrm{ln}\:{a}\right)\:{e}^{−\frac{{x}}{{a}}\:\mathrm{ln}\:{a}} =−\frac{\mathrm{ln}\:{a}}{{a}} \\ $$ $$\Rightarrow−\frac{{x}}{{a}}\:\mathrm{ln}\:{a}={W}\left(−\frac{\mathrm{ln}\:{a}}{{a}}\right) \\ $$ $$\Rightarrow{x}=−\frac{{a}}{\mathrm{ln}\:{a}}\:{W}\left(−\frac{\mathrm{ln}\:{a}}{{a}}\right)=−\frac{{W}\left(−\mathrm{ln}\:{a}^{\frac{\mathrm{1}}{{a}}} \right)}{\mathrm{ln}\:{a}^{\frac{\mathrm{1}}{{a}}} } \\ $$ $${if}\:{a}={e}:\:{one}\:{solution} \\ $$ $${if}\:{a}\neq{e}:\:{two}\:{solutions} \\ $$ | ||
Answered by Tinkutara last updated on 06/Apr/18 | ||
$${Consider}\:{f}\left({x}\right)={y}={x}^{\frac{\mathrm{1}}{{x}}} \\ $$ $$\frac{{dy}}{{dx}}=\frac{{y}}{{x}^{\mathrm{2}} }\left(\mathrm{1}−\mathrm{ln}\:{x}\right) \\ $$ $${If}\:{x}<{e},\:{f}\left({x}\right)\:{is}\:{increasing}. \\ $$ $${If}\:{x}>{e},\:{f}\left({x}\right)\:{is}\:{decreasing}. \\ $$ $$\therefore\:{e}^{\frac{\mathrm{1}}{{e}}} >\pi^{\frac{\mathrm{1}}{\pi}} \\ $$ $${e}^{\pi} >\pi^{{e}} \\ $$ | ||
Answered by artibunja last updated on 06/Apr/18 | ||