Question

把函數(shù)f(x)=sin²x-2sinxcosx+3cos²x(x∈R)的圖象沿x軸向左平移m個(gè)單位(m＞0)所得函數(shù)的圖象關(guān)于直線x=π對(duì)稱.

(1)求m的最小值.

(2)證明：當(dāng)x∈(π,π)時(shí)，經(jīng)過函數(shù)y=f(x)的圖象上任意兩點(diǎn)的直線斜率為負(fù)數(shù).

Answer 1

答案：(1)解：f(x)=cos2x-sin2x+2=cos(2x+)+2,將f(x)的圖象向左平移m個(gè)單位得函數(shù)g(x)=cos(2x+2m+)+2,其對(duì)稱軸x=π，

∴2×π+2m+=kπ(k∈Z).又m＞0，∴m_min=.                                

(2)證明：∵π＜x＜π,∴-4π＜2x+π，∴f(x)在(π,π)上為減函數(shù).

設(shè)x₁,x₂∈(π,π),且x₁＜x₂,則f(x₁)＞f(x₂),

∴k=＜0(用導(dǎo)數(shù)方法證明也可).