Proving that $8^x+4^x\geq 5^x+6^x$ for $x\geq 0$.

Use: $$ 4^x \left(\left(\frac 32\right)^x-1\right) \left(\left(\frac 43\right)^x-1\right) \ge 0 $$ for $x\ge 0$.

Hint. Let $f(t)=t^x$ then by the Mean Value Theorem there is $t_1\in (6,8)$ such that $$f(8)-f(6)=f'(t_1)(8-6)\Leftrightarrow 8^x-6^x=2xt_1^{x-1}.$$ Similarly there is $t_2\in (4,5)$ such that $$f(5)-f(4)=f'(t_2)(5-4)\Leftrightarrow 5^x-4^x=xt_2^{x-1}.$$ It remains to show that for $x\geq 0$ $$2xt_1^{x-1}\geq xt_2^{x-1}.$$

If $m$ and $n$ are integers, show that $\left|\sqrt{3}-\frac{m}{n}\right| \ge \frac{1}{5n^{2}}$

Can this be proved using Combinatorics or generating functions?

Why is square root by long division found so?

Why can we linearize the exponential regression?

$\int_0^1\frac{\ln x\ln^2(1-x^2)}{\sqrt{1-x^2}}dx=\frac{\pi}{2}\zeta(3)-2\pi\ln^32$

The Multiplication Operator $M_f: L^2(\mu) \to L^2(\mu)$ such that $M_f g = fg$ (Rudin)

Prove $\int_{\mathbb{R}^n}e^{-\max\{|x_1|,\ldots,|x_n|\}}dx=2^nn!$

Epic morphisms in the category of vector spaces. Is AC needed?

Characterization of nonabelian group $G$ such that for all $x,y\in G$, $xy\neq yx\implies x^2=y^2$.

Definition of a set

$\frac{\mathrm d^2 \log(\Gamma (z))}{\mathrm dz^2} = \sum\limits_{n = 0}^{\infty} \frac{1}{(z+n)^2}$

Ranges and the Fundamental Theorem of Calculus 1