TITLE: find the largest number
QUESTION [6 upvotes]: I have a question regarding this problem that is to find the largest number. Just by looking at the problem, I know the answer should be (d), but how can I prove that (d) is larger than (c) in a simple way?

Which of the following is the largest number:
$$\begin{align}
  &\textrm{(a)}\quad 3.14^3   
  &\quad\textrm{(b)}\quad 3^{3.14} 
  &\quad\textrm{(c)}\quad 3.14^{3.14} 
  &\quad\textrm{(d)}\quad (1/3)^{-4} 
\end{align}$$

REPLY [8 votes]: mathlove's solution is good. I think the following is a bit easier:
$3.14^{3.14} = 3^3\times\left(\frac{3.14}{3}\right)^3\times3.14^{0.14}$. So we just need the product of the second and third factors to be no more than 3.
The second factor is $<1.1^3=1.331<1.5$. The third factor is (much) less than $4^{0.5}=2$. So the product of these factors is $<3$.