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Câu 20 trang 226 SGK Đại số và Giải tích 11 Nâng cao

Chứng minh rằng phương trình
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Đề bài

Chứng minh rằng phương trình \({x^3} + a{x^2} + bx + c = 0\) luôn có ít nhất một nghiệm.

Lời giải chi tiết

Đặt \(f(x)={x^3} + a{x^2} + bx + c = 0\)

Do \(\mathop {\lim }\limits_{x \to  - \infty } f\left( x \right) =  - \infty \) nên  có số \(α < 0\) sao cho \(f(α) < 0\).Do \(\mathop {\lim }\limits_{x \to  + \infty } f\left( x \right) =  + \infty \) nên có số \(β > 0\) sao cho \(f(β) > 0\).Hàm số \(f\left( x \right) = {x^3} + a{x^2} + bx + c\) liên tục trên \(\mathbb R\) chứa đoạn \(\left[ {\alpha ;\beta } \right]\) nên theo định lý về giá trị trung gian của hàm số liên tục, tồn tại số \(d \in \left[ {\alpha ;\beta } \right]\) sao cho \(f(d) = 0\). Đó chính là nghiệm của phương trình \(f(x) = 0\).

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