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analysis90

analysis90

Đăng ký: 14-08-2010
Offline Đăng nhập: 02-09-2012 - 00:28
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Đề thi ôn tập thường xuyên của ĐHĐT

30-03-2012 - 09:46

MATHEMATICAL OLYMPIAD STUDENT

(third-2012)

Exercise 1. For $f(x)=2(x-1)-\arctan x,x\in\mathbb{R}$.
a) Prove that $f(x)=0$ have only a root $a\in(1,\sqrt{3})$.
b) Let $\{u_{n}\}_{n=1}^{\infty}$ be a sequence defined by$\left\{\begin{matrix} u_1=\dfrac{3}{2}& \\ u_n=1+\dfrac{1}{2}\arctan x,&n\geq1 \end{matrix}\right.$.
Prove that $\{u_{n}\}_{n=1}^{\infty}$ converges to $a$.
Exercise 2. Let $f:[0,+\infty)\longrightarrow \mathbb{R}$ be a differentiable function such that $f(0)=1$. Prove that if $f'(x)\geq f(x)$ for all $x\in[0,+\infty)$, then the function $g(x)=f(x)-e^x$ is a increasing function.
Exercise 3. Let $f:[0,1]\longrightarrow \mathbb{R}$ be a integrable function such that $\int_0^1xf(x)dx=0$. Prove that
$\int_0^1f^2(x)dx\geq 4(\int_0^1f(x)dx)^2$.
Exercise 4. Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ be a twice differentiable, $g:\mathbb{R}\longrightarrow \mathbb{R}^+$ be a function such that $f(x)+f''(x)=-xg(x)f'(x)$ for all $x\in\mathbb{R}$. Prove that $f(x)$ is bounded.
Exercise 5. Let $P(x)=\sum_{i=0}^{n}a_ix^i$ with $a_n>0$ be a $n$ degrees polynomial and have distinct $n$ roots. Prove that the polynomial $Q(x)=(P(x))^2-P'(x)$ only have
a) distinct $n+1$ roots if $n$ is odd.
b) distinct $n$ roots if $n$ is even.
Exercise 6. Find al function $f:\mathbb{R}\longrightarrow \mathbb{R}$ satisfies
$f(x+y)\geq f(x).f(y)\geq e^{x+y}$ for all $x,y\in \mathbb{R}$.

AB+BC+CA lớn nhất

16-03-2012 - 23:29

Cho trước một đường tròn $(O,R)$. Tìm 3 điểm A,B,C trên đường tròn sao cho AB+BC+CA lớn nhất.

mot cuon sach hay gianh cho sv nam hai day

21-08-2010 - 21:37

gtnb gianh cho nam hai day.sach ua vien toan hoc.

mot cuon sach hay ve giai tich co dien vien toan hoc day

21-08-2010 - 21:13

mot quyen sach hay cua vien gianh cho nhung ai sinh vien nam nhat

ai muon phieu luu voi bat dang thuc day?

21-08-2010 - 21:05

day la tai lieu kha hay ve bat dang thuc dac biet la nhung nguoi co tham vong muon chinh phuc bat dang thuc.