Question

If $a={\pi }^{3e},b=3^{\pi e}andc=e^{3\pi },then$

• A. $a<b<c$
• B. $b<a<c$
• C. $c<a<b$
• D. $a<c<b$

Answer 1

The correct option is A

$a<b<c$

Let $f\left(x\right)=x^{1/x},x>0logf\left(x\right)=\frac{1}{x}logx\Rightarrow \frac{f^{\prime }\left(x\right)}{f\left(x\right)}=\frac{1}{x^2}-\frac{1}{x^2}logx=\frac{1-logx}{x^2}\Rightarrow f^{\prime }\left(x\right)=\frac{1}{x^2}f\left(x\right)(1-logx)<0ifx>eThus,f\left(x\right)decreaseson[e,\infty ].Ase<3<\pi ,wegetf\left(\pi \right)<f\left(3\right)<f\left(e\right)\Rightarrow {\pi }^{1/\pi }<3^{1/3}<e^{1/e}\Rightarrow {\pi }^{3e}<3^{\pi e}<e^{3\pi }ora<b<c.$