How To Compare Transcendentals?

Use the graph of x^(1/x). You will see e^(1/e)>3^(1/3)>π^(1/π) then raise their power by 3eπ to get e^3π>3^eπ>π^3e.

---

Further elaboration :

We start with the the graph $y = x^{1/x}$ . I will prove that $y$ is a decreasing function for $x>e$ via derivatives.

We start by applying the logarithmic differentiation, $\ln y = \dfrac{\ln x}x$ . We differentiate the previous equation with respect to $x$ ,

$\dfrac 1y \cdot \dfrac{dy}{dx} = \dfrac{1 - \ln x}{x^2} \; .$

Since $y> 0$ for all $x>e$ and $1 - \ln x < 0$ for $x > e$ , then

Since for all $x>e$ , we know that $y>0$ and $1-\ln x< 0$ , then $RHS< 0$ and so $\dfrac{dy}{dx} < 0$ . Hence, $f(x) := y = x^{1/x}$ is a decreasing function for $x>e$ .

Because $e < 3 < \pi$ , then $f(e) > f(3) > f(\pi)$ . And so

$e^{1/e} > 3^{1/3} > \pi^{1/\pi} ,$

we raise the entire inequality by $3e\pi$ to obtain the desired answer $\large \boxed{ e^{3\pi} > 3^{\pi e} > \pi^{3e }}$ .

Note that $\frac{e+\pi}{2} \approx 3$ . Thus, we substitute $\pi=3+\delta$ and $e=3-\delta$ . We have

$\begin{aligned} 3^{\pi e} &\approx 3^{(3-\delta)(3+\delta)}\\ &= 3^{9-\delta^2} \end{aligned}$

Since $\delta < 1$ and is close to $0$ , we can substitute $\delta^2 \approx 0$ , so $3^{\pi e} \approx 3^9 = 19683$ . For $e^{3\pi}$ , we have

$\begin{aligned} e^{3\pi} &= (3-\delta)^{3(3+\delta)}\\ &=3^{9+3\delta} (1-\frac{\delta}{3})^{9+3\delta} \end{aligned}$

We expand to the zeroth order via Taylor's (because $\delta$ is reasonably small so that's ok)

$e^{3\pi} \approx 3^{9+3\delta} > 3^9 > 3^{\pi e}$

Doing the same thing with $\pi^{3e}$ , we get:

$π^{3e} = (3+\delta)^{3(3-\delta)} = 3^{9-3\delta} (1 + \frac{\delta}{3})^{9-3\delta} \approx 3^{9-3\delta}$

So we have:

$\begin{aligned} \pi^{3e} &\approx 3^{9-3\delta}\\ 3^{\pi e} &\approx 3^9\\ e^{3\pi} &\approx 3^{9+3\delta} \end{aligned}$

so:

$\pi^{3e} < 3^{\pi e} < e^{3\pi}$

---

Author's note: Of course this is just a fluke solution, as their differences are still fairly big and distinct. The error value for $3^{\pi e}$ is about 65%, so it is massive, but this functions for the question without using a calculator.