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Giải bài tập 1.33 trang 42 SGK Toán 12 tập 1 - Kết nối tri thức

Giá trị cực tiểu của hàm số (y = {x^2}ln x) là A. (frac{1}{e}). B. ( - frac{1}{e}). C. ( - frac{1}{{2e}}). D. (frac{1}{{2e}}).

♓Tổng hợp đề thi học kì 2 lớp 12 tất cả các môn - Kết nối tri thức

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

Giá trị cực tiểu của hàm số \(y = {x^2}\ln x\) là
A. \(\frac{1}{e}\).
B. \( - \frac{1}{e}\).
C. \( - \frac{1}{{2e}}\).
D. \(\frac{1}{{2e}}\).

Phương pháp giải - Xem chi tiết

Sử dụng kiến thức về cách tìm cực trị của hàm số \(y = f\left( x \right)\) để tìm cực tiểu của hàm số: 1. Tìm tập xác định của hàm số. 2. Tính đạo hàm f’(x). Tìm các điểm mà tại đó đạo hàm f’(x) bằng 0 hoặc đạo hàm không tồn tại. 3. Lập bảng biến thiên của hàm số. 4. Từ bảng biến thiên suy ra các cực tiểu của hàm số. 

Lời giải chi tiết

Tập xác định: \(D = \left( {0; + \infty } \right)\) Ta có: \(y' = 2x\ln x + \frac{{{x^2}}}{x} = 2x\ln x + x = x\left( {2\ln x + 1} \right)\) \(y' = 0 \Leftrightarrow x = \frac{1}{{\sqrt e }}\) (do \(x \in \left( {0; + \infty } \right)\)) Bảng biến thiên:

Do đó, chọn đáp án C

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