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Giải bài tập 1.42 trang 48 SGK Toán 12 tập 1 - Cùng khám phá

Cho hàm số \(y = \frac{{2x - 1}}{{x + 2}}\). Mệnh đề nào sau đây đúng? A. Hàm số đồng biến trên \(( - \infty ; - 2) \cup ( - 2; + \infty )\). B. Hàm số đồng biến trên \(( - \infty ;0)\). C. Hàm số đồng biến trên \(\mathbb{R}\). D. Hàm số đồng biến trên các khoảng \(( - \infty ; - 2)\) và \(( - 2; + \infty )\).
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Đề bài

Cho hàm số \(y = \frac{{2x - 1}}{{x + 2}}\). Mệnh đề nào sau đây đúng? A. Hàm số đồng biến trên \(( - \infty ; - 2) \cup ( - 2; + \infty )\). B. Hàm số đồng biến trên \(( - \infty ;0)\). C. Hàm số đồng biến trên \(\mathbb{R}\). D. Hàm số đồng biến trên các khoảng \(( - \infty ; - 2)\) và \(( - 2; + \infty )\).

Phương pháp giải - Xem chi tiết

- Tìm đạo hàm của hàm số - Xét dấu của đạo hàm để xác định chiều biến thiên của hàm số.

Lời giải chi tiết

Có thể loại đáp án C vì tập xác định của hàm số là \(D = R\backslash \{  - 2\} \) nên không thể đồng biến trên R. Đạo hàm của hàm số: \(y' = \frac{3}{{{{(x + 2)}^2}}} > 0\forall x \in D\). Dấu của y′ cho thấy rằng hàm số luôn đồng biến trên các khoảng mà nó xác định, tức là trên các khoảng (−∞,−2) và (−2,∞). Nhìn qua có thể thấy đáp án A và đáp D đều đúng nhưng cách diễn đạt của đáp án A là không hợp lý → Chọn D.

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