Question

把函數f(x)=sin²x-2sinxcosx+3cos²x的圖象沿x軸向左平移M個單位(M＞0),所得函數的圖象關于直線x=對稱.

(1)求M的最小值;

(2)證明當x∈(-,-)時,經過函數f(x)圖象上任意兩點的直線斜率恒為負數.

Answer 1

(1)解:f(x)=cos2x-sin2x+2?

=cos(2x+)+2.                                                                                                ?

將f(x)的圖象向左平移M個單位得函數g(x)=cos(2x+2M+)+2.                ?

由g(x)的對稱軸為x=,?

∴2×+2M+=kπ(k∈Z).又M＞0,?

∴M的最小值為.                                                                                                ?

(2)證明:∵-＜x＜-,?

∴-4π＜2x+＜-.?

∴f(x)在(-,-)上為減函數.                                                                  ?

設x₁,x₂∈(-,-),且x₁＜x₂,則f(x₁)＞f(x₂).?

∴k=＜0.