This question already has an answer here: Showing that for $n\geq 3$ the inequality $(n+1)^n<n^{(n+1)}$ holds 7 answers

Prove that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator. I did in the following way. Are there other ways? Proof : Let $f(x)=e\pi\frac{\ln x}{x}$. Then, $$e^{\pi}-{\pi}^e=e^{f(e)}-{e}^{f(\pi)}\tag1$$ Now, $$f'(x)=\frac{e\pi(1-\ln x)}{x^2},\quad f”(x)=\frac{e\pi (2\ln x-3)}{x^3},\quad f”'(x)=\frac{e\pi (11-6\ln x)}{x^4}.$$ Since $f'(x)\lt 0$ for $e\lt x\lt\pi$, one has $f(e)\gt f(\pi)$. By Taylor’s theorem, there exists a point $c$ in $(e,\pi)$ such […]

Someone asked me this question, and it bothers the hell out of me that I can’t prove either way. I’ve sort of come to the conclusion that 20! must be larger, because it has 36 prime factors, some of which are significantly larger than 2, whereas $2^{40}$ has only factors of 2. Is there a […]

show that $$\sqrt{7}^{\sqrt{8}}>\sqrt{8}^{\sqrt{7}}$$ and I found $$LHs-RHS=0.017\cdots$$ I have post this interesting problem Prove $\left(\frac{2}{5}\right)^{\frac{2}{5}}<\ln{2}$ can someone suggest any other nice method? Thank you everyone.

In this problem, I guess b is larger, but not know how to prove it without going to lengthy calculations. It is highly appreciated if anyone can give me a help. Which number is larger $$\begin{align} &\textrm{(a)}\quad 7^{94} &\quad\textrm{(b)}\quad 9^{91} \end{align}$$

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