函数y=4sin的平方x+4cosx+5【-π/3小于等于x小于等于（2π）/3】的值域是多少

函数y=4sin的平方x+4cosx+5【-π/3小于等于x小于等于（2π）/3】的值域是多少
函数y=4sin的平方x+4cosx+5【-π/3小于等于x小于等于（2π）/3】的值域是多少

函数y=4sin的平方x+4cosx+5【-π/3小于等于x小于等于（2π）/3】的值域是多少
y=4sin的平方x+4cosx+5
= 4(1-cos的平方x)+4cosx+5
= -4cos的平方x+4cosx+9
= -4(cosx-1/2)^2+10
-π/3小于等于x小于等于（2π）/3
-1/2≤cosx≤1
-1≤cosx-1/2≤1/2
0≤|cosx-1/2|≤1
-4≤-4(cosx-1/2)^2≤0
6≤-4(cosx-1/2)^2+10≤10
值域【6,10】

y＝4sin²x＋4cosx＋5＝4(1－cos²x)＋4cosx＋5＝﹣4cos²x＋4cosx＋9＝﹣4(cosx－1)²＋13
∵﹣π/3≤x≤2π/3 ∴﹣1/2≤cosx≤1
∴当cosx＝1时，y有最大值13
当cosx＝﹣1/2时，y有最小值4
∴值域是[4，13]

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