fx=sin^2x-cosx+1 (-π/2≤x≤π/2 ) 求值

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fx=sin^2x-cosx+1(-π/2≤x≤π/2)求值fx=sin^2x-cosx+1(-π/2≤x≤π/2)求值fx=sin^2x-cosx+1(-π/2≤x≤π/2)求值f(x)=sin&#

fx=sin^2x-cosx+1 (-π/2≤x≤π/2 ) 求值
fx=sin^2x-cosx+1 (-π/2≤x≤π/2 ) 求值

fx=sin^2x-cosx+1 (-π/2≤x≤π/2 ) 求值
f(x)=sin²x-cosx+1
=1-cos²x-cosx+1
=-cos²x-cosx
=-(cosx +1/2)² +1/4
-π/2≤x≤π/2 0≤cosx≤1 1/2≤cosx+1/2≤3/2
1/4≤(cosx+1/2)²≤9/4 -9/4≤-(cosx+1/2)²≤-1/4
-2≤-(cosx+1/2)²+1/4≤0
-2≤f(x)≤0,函数的值域为[-2,0],最大值为0,最小值为-2

fx=sin^2x-cosx+1=1-cos^2x-cosx+1=-(cosx+1/2)^2+2+1/4=-(cosx+1/2)^2+9/4,因为(-π/2≤x≤π/2)所以cosx∈[-1,1],所以f(x)max=-(-1/2+1/2)^2+9/4=9/4,f(x)min=-(1+1/2)^2+9/4=0