Giải bài 3 trang 31 sách bài tập toán 12 - Chân trời sáng tạo

Cho hàm số $y = 2{x^3} + 6{x^2} - x + 2$. Viết phương trình tiếp tuyến của đồ thị hàm số tại tâm đối xứng của nó.

📖Tổng hợp đề thi học kì 2 lớp 12 tất cả các môn - Chân trời sáng tạo

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Đề bài

Cho hàm số $y = 2{x^3} + 6{x^2} - x + 2$. Viết phương trình tiếp tuyến của đồ thị hàm số tại tâm đối xứng của nó.

Phương pháp giải - Xem chi tiết

‒ Hoành độ tâm đối xứng là nghiệm của phương trình $y''=0$. ‒ Phương trình tiếp tuyến của đồ thị hàm số $y = f\left( x \right)$ tại điểm ${M_0}\left( {{x_0};{y_0}} \right)$: $y = f'\left( {{x_0}} \right)\left( {x - {x_0}} \right) + {y_0}$

Lời giải chi tiết

$y'=6{{x}^{2}}+12x-1;y''=12x+12;y''=0\Leftrightarrow x=-1$ Tâm đối xứng $I$ của đồ thị hàm số có toạ độ $\left( { - 1;7} \right)$. Ta có $y'\left( { - 1} \right) = - 7$. Phương trình tiếp tuyến của đồ thị hàm số tại $I\left( { - 1;7} \right)$: $y = - 7\left( {x + 1} \right) + 7$ hay $y = - 7x$.

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4.9 trên 7 phiếu

▨ Giải bài 4 trang 31 sách bài tập toán 12 - Chân trời sáng tạo Với giá trị nào của $m$ thì đồ thị của hàm số $y = - {x^3} - 3{x^2} + mx + 1$ có tâm đối xứng nằm trên trục $Ox$? Khi đó, có thể kết luận gì về số giao điểm của đồ thị hàm số với trục hoành?
♛ Giải bài 5 trang 31 sách bài tập toán 12 - Chân trời sáng tạo Khảo sát và vẽ đồ thị của các hàm số sau: a) (y = 3 + frac{1}{x}); b) (y = 2 - frac{1}{{1 + x}}).
𝄹 Giải bài 6 trang 32 sách bài tập toán 12 - Chân trời sáng tạo Ta đã biết đồ thị hàm số $y = \frac{{2{\rm{x}} - 1}}{{x + 1}}$ có tiệm cận đứng là đường thẳng $x = - 1$ và tiệm cận ngang là đường thẳng $y = 2$. a) Tìm toạ độ giao điểm $I$ của đường tiệm cận. b) Với $t$ tuỳ ý $\left( {t \ne 0} \right)$, gọi $M$ và $M'$ lần lượt là hai điểm trên đồ thị hàm số có hoành độ lần lượt là ${x_M} = {x_I} - t$ và ${x_{M'}} = {x_I} + t$. Tìm các tung độ $y\left( {{x_M}} \right)$ và $y\left( {{x_{M'}}} \right)$. Từ đó, chứng minh rằng hai đ
🗹 Giải bài 7 trang 32 sách bài tập toán 12 - Chân trời sáng tạo Cho hàm số $y = \frac{{2{\rm{x}} - 1}}{{ - x + 3}}$. Chứng tỏ rằng đường thẳng $y = - x$ cắt đồ thị hàm số đã cho tại hai điểm phân biệt.
ꦛ Giải bài 8 trang 32 sách bài tập toán 12 - Chân trời sáng tạo Khảo sát và vẽ đồ thị của các hàm số sau: a) (y = frac{{{x^2} - 2{rm{x}} + 2}}{{{rm{x}} - 1}}); b) (y = - 2{rm{x}} + frac{1}{{2{rm{x}} + 1}}).

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Luyện Bài Tập Trắc nghiệm Toán 12 - Chân trời sáng tạo - Xem ngay

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