数学全国100所名校单元测试示范卷·数学卷（七）
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1、
[f^(π/4－x)＋f^(π/4＋x)]/2
=[cos^(π/4－x)＋cos^(π/4＋x)]/2
=[(cos(π/2-2x)＋1)/2＋ (cos(π/2＋2x)＋1)/2]/2
=[sin2x＋1＋ (-sin2x＋1)]/4
=2/4
=1/2
f^(π/4－x)f^(π/4＋x)
=cos^(π/4－x)cos^(π/4＋x)
=[(cos(π/2-2x)+1]/2×[(cos(π/2+2x)＋1]/2
=(1+sin2x)(1-sin2x)/4
=(1-sin^2x)/4
=cos^2x/4≤1/4
[f^(π/4－x)＋f^(π/4＋x)]/2
=[cos^(π/4－x)＋cos^(π/4＋x)]/2
=[(cos(π/2-2x)＋1)/2＋ (cos(π/2＋2x)＋1)/2]/2
=[sin2x＋1＋ (-sin2x＋1)]/4
=2/4
=1/2
f^(π/4－x)f^(π/4＋x)
=cos^(π/4－x)cos^(π/4＋x)
=[(cos(π/2-2x)+1]/2×[(cos(π/2+2x)＋1]/2
=(1+sin2x)(1-sin2x)/4
=(1-sin^2x)/4
=cos^2x/4≤1/4
追答
所以 [f^(π/4－x)＋f^(π/4＋x)]/2≥√f^(π/4－x)f^(π/4＋x)
2、4f(π/3－x)f(x)f(π/3+ x)
=4cos(π/3-x)cosxcos(π/3+x)
=4cosx(cos2x＋ cos2π/3)/2
=2cosxcos2x ＋2cos2π/3cosx
=(cos3x＋ cosx) ＋2×(-1/2)×cosx
=cos3x＋cosx－cosx
=cos3x
=f(3x)
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