# Ngô Quốc Anh

## September 23, 2007

### “Hàm số” vs. “Số phức”

Filed under: Các Bài Tập Nhỏ — Ngô Quốc Anh @ 18:27

1) Let $z_{1},z_{2}\in\mathbb{C}$ such as $z_{1}= f(a)+if(b)$, $z_{2}= f(b)-f(a)i$ and also $\parallel z_{1}|-|z_{2}\parallel = |z_{1}|+|z_{2}|$, where $f$ is a differentiable function at $[a,b]$. Prove that exists $\xi_{1},\xi_{2}\in(a,b)$ such as $f'(\xi_{1})+f'(\xi_{2}) = 0$.

2) Let $f$ continuous function to $[a,b]$ such as $f(x)\neq0$ $\forall x\in[a,b]$. Also let $z\in\mathbb{C}$ such as $z+\frac{1}{z}= f(a)$ and $z^{2}+\frac{1}{z^{2}}= f^{2}(b)$. Prove that $i)|f(a)| > |f(b)|$ and $ii)$ the equation $f(b)+x^{3}f(a) = 0$ has at least one real root to $(-1,1)$.

3) Let $f$ differentiable function to $[a,b]$. Also let $z,w\in\mathbb{C}$ such as $|z-iw|^{2}= |z|^{2}+|iw|^{2}$ and $f(b)\neq0$ . Prove that exists $\xi\in (a,b)$ such as $\xi f'(\xi) = f(\xi)$.

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