Phần I: Lý thuyết
http://dientuvietnam.net/cgi-bin/mimetex.cgi?y'=\dfrac{x^2-2x-m}{(x-1)^2}=0
Hàm số có cực trị khi và chỉ khi '=1+m>0 hay m>-1
Khi đó:
http://dientuvietnam.net/cgi-bin/mimetex.cgi?x_{max}+x_{min}=2
http://dientuvietnam.net/cgi-bin/mimetex.cgi?x_{max}x_{min}=-m
http://dientuvietnam.net/cgi-bin/mimetex.cgi?\dfrac{1}{x_{max}}+\dfrac{1}{x_{min}}=\dfrac{-2}{m}
http://dientuvietnam.net/cgi-bin/mimetex.cgi?\dfrac{1}{x_{max}}\dfrac{1}{x_{min}}=\dfrac{-1}{m}
Ta có http://dientuvietnam...mimetex.cgi?x_o là hoành độ cực trị thì http://dientuvietnam...metex.cgi?y(x_o)=\dfrac{u'(x_o)}{v'(x_o)}=2x_o-2m (y=2x-2m là đường thẳng đi qua 2 cực trị)
Từ đó:
http://dientuvietnam.net/cgi-bin/mimetex.cgi?y_{max}=2x_{max}-2m
http://dientuvietnam.net/cgi-bin/mimetex.cgi?y_{min}=2x_{min}-2m
http://dientuvietnam.net/cgi-bin/mimetex.cgi?y_{max}+y_{min}=4-4m
http://dientuvietnam.net/cgi-bin/mimetex.cgi?y_{max}y_{min}=4x_{max}x_{min}-4(x_{max}+x_{min})+4m^2=4m^2-12m
Từ http://dientuvietnam.net/cgi-bin/mimetex.cgi?\dfrac{y_{max}}{x_{max}}=2-\dfrac{2m}{x_max}
Suy ra:
http://dientuvietnam.net/cgi-bin/mimetex.cgi?\dfrac{y_{max}}{x_{max}}+\dfrac{y_{min}}{x_{min}}=4-2m(\dfrac{1}{x_{max}}+\dfrac{1}{x_{min}})=8
Tương tự http://dientuvietnam.net/cgi-bin/mimetex.cgi?\dfrac{y_{max}}{x_{max}}.\dfrac{y_{min}}{x_{min}}=-4m+12
Phần 2: Bài tập áp dụng
Tìm m để:
1) http://dientuvietnam.net/cgi-bin/mimetex.cgi?|x_{max}-x_{min}|=2 ( Hay hai cự trị có khoảng cách trên trục hoành có độ dài bằng 2)
2) http://dientuvietnam.net/cgi-bin/mimetex.cgi?x_{max}^3+x_{min}^3=8
3) http://dientuvietnam.net/cgi-bin/mimetex.cgi?x_{max}<x_{min}<2
4) http://dientuvietnam.net/cgi-bin/mimetex.cgi?3x_{max}+x_{min}=2
5) http://dientuvietnam.net/cgi-bin/mimetex.cgi?|\dfrac{1}{x_{max}}-\dfrac{1}{x_{min}}|=2
6) http://dientuvietnam.net/cgi-bin/mimetex.cgi?\dfrac{1}{x_{max}^2}+\dfrac{1}{x_{min}^2}=4
7) http://dientuvietnam.net/cgi-bin/mimetex.cgi?\dfrac{1}{x_{max}}<\dfrac{1}{x_{min}}<2
8) http://dientuvietnam.net/cgi-bin/mimetex.cgi?\dfrac{2}{x_{max}}+\dfrac{1}{x_{min}}=3
9) http://dientuvietnam.net/cgi-bin/mimetex.cgi?y_{max}y_{min}<0
10)http://dientuvietnam.net/cgi-bin/mimetex.cgi?|y_{max}-y_{min}|=4
11)http://dientuvietnam.net/cgi-bin/mimetex.cgi?y_{max}<y_{min}<4
12)http://dientuvietnam.net/cgi-bin/mimetex.cgi?y_{max}^2+y_{min}^2=3
13)http://dientuvietnam.net/cgi-bin/mimetex.cgi?y_{max}+2y_{min}=3
14)http://dientuvietnam.net/cgi-bin/mimetex.cgi?|\dfrac{y_{max}}{x_{max}}-\dfrac{y_{min}}{x_{min}}|=2
15)http://dientuvietnam.net/cgi-bin/mimetex.cgi?|\dfrac{x_{max}}{y_{max}}-\dfrac{x_{min}}{y_{min}}|=2
16)http://dientuvietnam.net/cgi-bin/mimetex.cgi?(\dfrac{y_{max}}{x_{max}})^3+(\dfrac{y_{min}}{x_{min}})^3=8
17)http://dientuvietnam.net/cgi-bin/mimetex.cgi?(\dfrac{x_{max}}{y_{max}})^2+(\dfrac{x_{min}}{y_{min}})^2=8
18)http://dientuvietnam.net/cgi-bin/mimetex.cgi?3\dfrac{x_{max}}{y_{max}}+\dfrac{x_{min}}{y_{min}}|=6
19)http://dientuvietnam.net/cgi-bin/mimetex.cgi?3\dfrac{y_{max}}{x_{max}}+\dfrac{y_{min}}{x_{min}}=4
20)http://dientuvietnam.net/cgi-bin/mimetex.cgi?4y_{max}+6y_{min}+x_{max}+3x_{min}=7
Bài viết đã được chỉnh sửa nội dung bởi hoacomay: 18-06-2005 - 07:00