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

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 $$\begin{align} &\textrm{(a)}\quad 7^{94} &\quad\textrm{(b)}\quad 9^{91} \end{align}$$

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.

Intereting Posts

Derivation of the polarization identities?
How to go from beginner to expert in mathematics?
Online tool for making graphs (vertices and edges)?
What is the fraction field of $R]$, the power series over some integral domain?
About the Legendre differential equation
Primes for which $x^k\equiv n\pmod p$ is solvable: the fixed version
how could we compute this infinite real integral using complex methods?
Hermitian matrix such that $4M^5+2M^3+M=7I_n$
Defining the multidimensional Riemann Integral as a limit of certain sums
Find $\det X$ if $8GX=XX^T$
The Eigenvalues of a block matrix
In which topological spaces is every singleton set a zero set?
Adjoint functors
Why is $\pi^2$ so close to $10$?
Sum identity using Stirling numbers of the second kind