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Giải bài 35 trang 78 sách bài tập toán 11 - Cánh diều

Cho hàm số \(f\left( x \right) = {x^3} + 4{x^2} + 5.\) Giải bất phương trình

⛄Tổng hợp đề thi học kì 2 lớp 11 tất cả các môn - Cánh diều

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

Cho hàm số \(f\left( x \right) = {x^3} + 4{x^2} + 5.\) Giải bất phương trình

\(f'\left( x \right) - f''\left( x \right) \ge 0.\)

Phương pháp giải - Xem chi tiết

Tính \(f'\left( x \right),{\rm{ }}f''\left( x \right)\) để giải bất phương trình .

Lời giải chi tiết

\(f\left( x \right) = {x^3} + 4{x^2} + 5 \Rightarrow f'\left( x \right) = 3{x^2} + 8x \Rightarrow f''\left( x \right) = 6x + 8.\)Theo đề bài: \(f'\left( x \right) - f''\left( x \right) \ge 0 \Leftrightarrow 3{x^2} + 8x - \left( {6x + 8} \right) \ge 0 \Leftrightarrow 3{x^2} + 2x - 8 \ge 0\)\(\left( {3x - 4} \right)\left( {x + 2} \right) \ge 0 \Leftrightarrow \left[ \begin{array}{l}x \ge \frac{4}{3}\\x \le  - 2\end{array} \right.\)Tập nghiệm của bất phương trình là: \(\left( { - \infty ; - 2} \right] \cup \left[ {\frac{4}{3}; + \infty } \right).\)

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