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Giải bài 10 trang 91 sách bài tập toán 11 - Chân trời sáng tạo tập 1

Chứng minh rằng phương trình: a) \({x^3} + 2x - 1 = 0\) có nghiệm thuộc khoảng \(\left( { - 1;1} \right)\); b) \(\sqrt {{x^2} + x} + {x^2} = 1\) có nghiệm thuộc khoảng \(\left( {0;1} \right)\).

💯Tổng hợp đề thi học kì 2 lớp 11 tất cả các môn - Chân trời sáng tạo

Toán - Văn - Anh - Lí - Hóa - Sinh
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Đề bài

Chứng minh rằng phương trình: a) \({x^3} + 2x - 1 = 0\) có nghiệm thuộc khoảng \(\left( { - 1;1} \right)\); b) \(\sqrt {{x^2} + x}  + {x^2} = 1\) có nghiệm thuộc khoảng \(\left( {0;1} \right)\).

Phương pháp giải - Xem chi tiết

Sử dụng kiến thức về ứng dụng tính liên tục của hàm số vào xét sự tồn tại nghiệm của phương trình để chứng minh: Nếu hàm số \(y = f\left( x \right)\) liên tục trên đoạn \(\left[ {a;b} \right]\) và \(f\left( a \right).f\left( b \right) < 0\) thì luôn tồn tại ít nhất một điểm \(c \in \left( {a;b} \right)\) sao cho \(f\left( c \right) = 0\).

Lời giải chi tiết

a) Xét hàm số \(f\left( x \right) = {x^3} + 2x - 1\), f(x) liên tục trên \(\left[ { - 1;1} \right]\) và có \(f\left( { - 1} \right) =  - 4,f\left( 1 \right) = 2\). Do \(f\left( { - 1} \right).f\left( 1 \right) < 0\) nên phương trình \({x^3} + 2x - 1 = 0\) có nghiệm thuộc khoảng \(\left( { - 1;1} \right)\). b) Xét hàm số \(f\left( x \right) = \sqrt {{x^2} + x}  + {x^2} - 1\), f(x) liên tục trên \(\left[ {0;1} \right]\) và có \(f\left( 0 \right) =  - 1,f\left( 1 \right) = \sqrt 2 \). Do \(f\left( 0 \right).f\left( 1 \right) < 0\) nên phương trình \(f\left( x \right) = 0\) hay phương trình \(\sqrt {{x^2} + x}  + {x^2} = 1\) có nghiệm thuộc khoảng \(\left( {0;1} \right)\).

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