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Câu 38 trang 213 SGK Đại số và Giải tích 11 Nâng cao

Cho hàm số
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Cho hàm số \(y = {\cos ^2}x + m\sin x\) (m là tham số) có đồ thị là (C). Tìm m trong mỗi trường hợp sau:

LG a

Tiếp tuyến của (C) tại điểm với hoành độ \(x = π\) có hệ số góc bằng 1.

Phương pháp giải:

Giải phương trình \(f'(\pi )=1\) tìm m.

Lời giải chi tiết:

Đặt \(f\left( x \right) = {\cos ^2}x + m\sin x,\) ta có : \(f'\left( x \right) = 2\cos x\left( { - \sin x} \right) + m\cos x\) \(=  - \sin 2x + m\cos x\) Hệ số góc tiếp tuyến của (C) tại điểm có hoành độ \(x = π\) là : \(\eqalign{  & f'\left( \pi  \right) =  - \sin 2\pi  + m\cos \pi  =  - m  \cr  & \text{Vậy}\,f'\left( \pi  \right) = 1 \Leftrightarrow m =  - 1 \cr} \)

LG b

Hai tiếp tuyến của (C) tại các điểm có hoành độ \(x =  - {\pi  \over 4}\)  và \(x = {\pi  \over 3}\) song song hoặc trùng nhau.

Phương pháp giải:

Giải phương trình \(f'\left( { - {\pi  \over 4}} \right) = f'\left( {{\pi  \over 3}} \right)\) tìm m.

Lời giải chi tiết:

Theo đề bài, ta có : \(\eqalign{  & f'\left( { - {\pi  \over 4}} \right) = f'\left( {{\pi  \over 3}} \right)  \cr  &  \Leftrightarrow  - \sin \left( { - {\pi  \over 2}} \right) + m\cos \left( { - {\pi  \over 4}} \right) \cr &=  - \sin {{2\pi } \over 3} + m\cos {\pi  \over 3}  \cr  &  \Leftrightarrow 1 + m{{\sqrt 2 } \over 2} =  - {{\sqrt 3 } \over 2} + {m \over 2} \cr &\Leftrightarrow m = {{\sqrt 3  + 2} \over {1 - \sqrt 2 }} \cr} \)

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