If $x=\pi^{3}$ and $y=3^{\pi}$ then what is the relation between x and y?
I have no idea of how to solve this problem. Also in general there is no relation between two numbers of this kind. So what is so special about this $\pi$ and 3.
Any help would be great, thank you.
First of all, consider the function $f(x)=x^\frac{1}{x}$.
Find the derivative of $f(x)=x^\frac{1}{x}$. You will get that the maxima is attained at $x=e$ and that the function is increasing for $x < e$ and decreasing for $x > e$.
Hence you have $x_2^\frac{1}{x_2}\le x_1^\frac{1}{x_1}$ $\forall$ $x_2> x_1 >e$ and $ \in \mathbb{R}$.
Now $\pi > 3 > e$
So we have $$\pi^\frac{1}{\pi} < 3^\frac{1}{3}$$ and hence your result $$\pi^3 < 3^\pi$$
By relation here, it is meant
Which relation holds: $\pi^3 < 3^\pi$, $\pi^3 = 3^\pi$, or $\pi^3 > 3^\pi$ ?
Rewrite this comparison to compare $\pi^{1/\pi}$ and $3^{1/3}$:
Which relation holds: $\pi^{1/\pi} < 3^{1/3}$, $\pi^{1/\pi} = 3^{1/3}$, or $\pi^{1/\pi} > 3^{1/3}$ ?
Then this is a classical problem, whose solution clearly now lies in considering the function $x^{1/x}$ and where it is monotone. You'll see that the only thing special about $\pi$ and $3$ is that $\pi > 3$.
