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Câu 53 trang 176 SGK Đại số và Giải tích 11 Nâng cao

Chứng minh rằng
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Đề bài

Chứng minh rằng phương trình \({x^3} + x + 1 = 0\) có ít nhất một nghiệm âm lớn hơn -1.

Phương pháp giải - Xem chi tiết

𓄧Sử dụng định lý: Nếu hàm số f(x) liên tục trên đoạn [a;b] và \(f(a).f(b)<0\) thì tồn tại ít nhất một điểm c∈(a;b) sao cho f(c)=0.

Lời giải chi tiết

Hàm số \(f\left( x \right) = {x^3} + x + 1\) liên tục trên đoạn [-1 ; 0] có \(f(-1) = -1\) và \(f(0) = 1\).Vì \(f(-1)f(0) < 0\) nên theo hệ quả của định lí về giá trị trung gian của hàm số liên tục, tồn tại ít nhất một điểm \(c \in (-1 ; 0)\) sao cho \(f(c) = 0\). Số c là nghiệm âm lớn hơn -1 của phương trình đã cho.

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