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Giải câu hỏi mở đầu trang 5 SGK Toán 12 tập 1 - Cánh diều

Một doanh nghiệp dự kiến lợi nhuận khi sản xuất x sản phẩm \((0 \le x \le 300)\) được cho bởi hàm số \(y = - {x^3} + 300{x^2}\) (đơn vị: nghìn đồng) và được minh họa bằng đồ thị ở Hình 1. Sự thay đổi lợi nhuận theo số sản phẩm sản xuất ra và dấu của đạo hàm y’ có mối liên hệ với nhau như thế nào?

ꦓTổng hợp đề thi học kì 2 lớp 12 tất cả các môn - Cánh diều

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

Một doanh nghiệp dự kiến lợi nhuận khi sản xuất x sản phẩm \((0 \le x \le 300)\) được cho bởi hàm số \(y =  - {x^3} + 300{x^2}\) (đơn vị: nghìn đồng) và được minh họa bằng đồ thị ở Hình 1. Sự thay đổi lợi nhuận theo số sản phẩm sản xuất ra và dấu của đạo hàm y’ có mối liên hệ với nhau như thế nào?

Phương pháp giải - Xem chi tiết

Tính đạo hàm y'. Nhận xét dấu của y' và sự thay đổi lợi nhuận (tăng hay giảm).

Lời giải chi tiết

Ta có \(y' =  - 3{x^2} + 600x = 0 \Leftrightarrow \left[ \begin{array}{l}x = 0\\x = 2\end{array} \right.\)

Bảng xét dấu của y' trên đoạn [0;300]:

Quan sát Hình 1, ta thấy lợi nhuận theo số sản phẩm tăng thì y’ mang dấu dương; lợi nhuận theo số sản phẩm giảm thì y’ mang dấu âm.

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