Có bao nhiêu giá trị nguyên dương của tham số \(m\) để hàm số \(y=\frac{8}{3} x^{3}+2 \ln x-m x\) đồng biến trên \((0 ; 1)\)?
Hãy suy nghĩ và trả lời câu hỏi trước khi xem đáp án
TXĐ \(D=\mathbb{R}\)
Ta có \(y^{\prime}=8 x^{2}+\frac{2}{x}-m\). Yêu cầu bài toán \(\Leftrightarrow y^{\prime} \geq 0 \forall x \in(0 ; 1)\)
\(\Leftrightarrow 8 x^{2}+\frac{2}{x}-m \geq 0 \forall x \in(0 ; 1) \Leftrightarrow h(x)=8 x^{2}+\frac{2}{x} \geq m \forall x \in(0 ; 1) \Leftrightarrow m \leq \min _{(0 ; 1)} h(x)\).
Xét hàm \(h(x)=8 x^{2}+\frac{2}{x} \forall x \in(0 ; 1)\). Ta có \(h^{\prime}(x)=16 x-\frac{2}{x^{2}} \Rightarrow h^{\prime}(x)=0 \Rightarrow x=\frac{1}{2}\).
Bảng biến thiên
Từ \(\mathrm{BBT} \Rightarrow m \leq 6\), kết hợp với \(m\) nguyên dương ta được \(m \in\{1 ; 2 ; 3 ; 4 ; 5 ; 6\}\).
Được thiết kế với định hướng phát triển năng lực toàn diện, Đề Thi Tham Khảo Đánh Giá Năng Lực Năm 2025 – ĐHQG Hà Nội – Đề Số 3 không chỉ kiểm tra kiến thức mà còn đo lường khả năng phân tích, lập luận và giải quyết vấn đề. Với ba phần thi: Toán Học Và Xử Lí Số Liệu, Văn Học - Ngôn Ngữ, và Khoa Học/Tiếng Anh, thí sinh được thử thách qua các dạng bài phong phú như trắc nghiệm, điền đáp án và giải quyết tình huống. Đặc biệt, phần Khoa Học cho phép thí sinh tự chọn 3 trong 5 chủ đề để thể hiện năng lực chuyên biệt của mình, tạo cơ hội tối ưu hóa điểm số theo thế mạnh cá nhân.
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