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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