Record of solved math202/3-level problems

首兩條的解題方法在 202 我們多次看到，作為回憶從某論壇 copy 下來。

Problem 1. Let $f:[a,+\infty)\to \mathbb{R}$ a twice differentiable function on the interval $(a,+\infty)$  with $\lim_{n\to\infty}f(x)=l\in\mathbb{R}$  and $|f''(x)|\leq M,\forall x \in [a,+\infty)$. Prove that $\displaystyle \lim_{x\to\infty}f'(x)=0$.

Problem 2. Let $f$ be differentiable on $(a,+\infty)$. If $\lim_{x\to\infty}\big(f(x)+xf'(x)\ln x\big)=l$, then show that $\lim_{x\to\infty}f(x)=l$.

以下這一條取自 2010 Department of Mathematics, Graduate School of Science, Kyoto University subjects of tests : Mathematics I

Problem 3*. Let $f$ be continuous real valued function on $[0,1]$ and $f(0)=0, f(1)=1$. Find $$\lim_{n\to\infty}n\int_0^1f(x)x^{2n}\,dx.$$
基本上 $f(0)=0, f(1)=1$ 是一多餘的 condition。把它們保留下來作為 unknown 可得到一般結果。我想多餘的資料旨在 confuse 解題的人。把 2 換成 $k$，結果應該差不多！

以下在 analysis section 裏看到這條，可能與 Lagrange multiplier 有關吧。其實只需中學知識。

Problem 4. Let $x_1,x_2,\dots,x_n$ be positive real number satisfying $\displaystyle \sum_{k=1}^n\frac{1}{x_k}=n$. Find the minimum of $$\sum_{k=1}^n\frac{(x_k)^k}{k}.$$
Problem 5. Find $\displaystyle\lim_{n\to\infty}\left\{n^{2}\left(\int_{0}^{1}\sqrt[n]{1+x^{n}}\,dx-1\right)\right\}$.

Problem 6. Suppose that $f:\mathbb{R}-\{0,1\}\to \mathbb{R}$ satisfies the equation
$\displaystyle f(x)+f\left(\frac{x-1}{x}\right)=1+x$, find $f(x)$.

Problem 7. Let $f:[0,1]\to\mathbb{R}$ be differentiable on $[0,1]$ with $f(1)=0$, show that
$\displaystyle \lim_{n\to+\infty}n^{2}\int_{0}^{1}f(x)x^{n}\, dx =-f'(1)$.


都是些 202 基本概念便能夠解決的問題。