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Giải bài tập 5 trang 37 SGK Toán 12 tập 1 - Chân trời sáng tạo

Giá trị nhỏ nhất của hàm số \(y = \sqrt {{x^2} + 2x + 3} \) trên đoạn [–2; 3] là A. \(\sqrt 3 \) B. \(\sqrt {30} \) C. \(\sqrt 2 \) D. 0

꧂Tổng hợp đề thi học kì 2 lớp 12 tất cả các môn - Chân trời sáng tạo

Toán - Văn - Anh - Hoá - Sinh - Sử - Địa
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Đề bài

 
 
Giá trị nhỏ nhất của hàm số \(y = \sqrt {{x^2} + 2x + 3} \) trên đoạn [–2; 3] là A. \(\sqrt 3 \)                   B. \(\sqrt {30} \)              C. \(\sqrt 2 \)                   D. 0
 

Phương pháp giải - Xem chi tiết

Cho hàm số y = f(x) xác định trên tập hợp D. - Số M được gọi là giá trị lớn nhất của hàm số y = f(x) trên D nếu f(x) \( \le \) M với mọi x thuộc D và tồn tại \({x_0}\) thuộc D sao cho f(\({x_0}\)) = M. Kí hiệu M = \(\mathop {\max }\limits_D \)f(x). - Số m được gọi là giá trị nhỏ nhất của hàm số y = f(x) trên D nếu f(x) \( \ge \) m với mọi x thuộc D và tồn tại \({x_0}\) thuộc D sao cho f(\({x_0}\)) = m. Kí hiệu m = \(\mathop {\min }\limits_D \)f(x).
 

Lời giải chi tiết

Chọn C Tập xác định: \(D = \mathbb{R}\) \(y' = \frac{{x + 1}}{{\sqrt {{x^2} + 2x + 3} }} = 0 \Leftrightarrow x =  - 1\) Bảng biến thiên:

Từ bảng biến thiên ta thấy, \(\mathop {\min }\limits_D y = y( - 1) = \sqrt 2 \)
 

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