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 $f\left(x\right)=x^{1/x},x>0logf\left(x\right)=\frac{1}{x}logx\Rightarrow \frac{f^{\prime }\left(x\right)}{f\left(x\right)}=\frac{1}{x^2}-\frac{1}{x^2}logx=\frac{1-logx}{x^2}\Rightarrow f^{\prime }\left(x\right)=\frac{1}{x^2}f\left(x\right)(1-logx)<0ifx>e$
Thus, f(x) decreases on $(e,\infty )$.
As e<3< $\pi$, we get
$f\left(\pi \right)<f\left(3\right)<f\left(e\right)\Rightarrow (\pi {)}^{1/\pi }<3^{1/3}<e^{1/e}\Rightarrow {\pi }^{3e}<3^{\pi e}<e^{3\pi }ora<b<c.$