已知:f(x)=-(sinx)^2+sinx+a
\u5df2\u77e5\u51fd\u6570f\uff08x\uff09=f\uff08\u03c0-x\uff09\uff0c\u4e14\u5f53x\u2208 \u65f6\uff0cf\uff08x\uff09=x+sinx\uff0c\u8bbea=f\uff081\uff09\uff0cb=f\uff082\uff09\uff0cc=f\uff083\uff09\uff0c\u5219 [D
\u89e3\uff1a
\uff081\uff09
f(x)=0\u6709\u5b9e\u6570\u89e3,
\u7b49\u4ef7\u4e8ea=sin^2x-sinx\u6709\u5b9e\u6570\u89e3,
\u5373a\u7684\u53d6\u503c\u5728\u51fd\u6570g(x)=sin^2x-sinx\u7684\u503c\u57df\u5185\u3002
g(x)= sin^2x-sinx
=(sinx-1/2)^2-1/4,
Sinx=1/2\u65f6\uff0cg(x)\u53d6\u5230\u6700\u5c0f\u503c-1/4.
Sinx=-1\u65f6\uff0cg(x)\u53d6\u5230\u6700\u5927\u503c2.
\u6240\u4ee5g(x)\u2208[-1/4,2]
\u2234f(x)=0\u6709\u5b9e\u6570\u89e3\u65f6\uff0ca\u7684\u53d6\u503c\u8303\u56f4\u662f[-1/4,2]\u3002
\uff082\uff09\u5f53x\u2208R\u65f6m=sinx\u2208[-1,1]
f(x)=-sin^2x+sinx+a=-m^2+m+a=-(m-1/2)^2+a+1/4
\u22351\u2264f(x)\u226417/4
\u22341\u2264-(m-1/2)^2+a+1/4\u226417/4
-17/4\u2264(m-1/2)^2-a-1/4\u2264-1
\u5f53m\u2208[-1,1]\u65f6(m-1/2)^2\u2208[0,9/4]
\u2234-a-1/4\u2264(m-1/2)^2-a-1/4\u22649/4-a-1/4
\u2235\u5fc5\u987b\u6ee1\u8db3-17/4\u2264(m-1/2)^2-a-1/4\u2264-1
\u2234-a-1/4\u2265-1\u4e149/4-a-1/4\u2264-17/4
\u2234 4\u2264a\u22649
a\u7684\u8303\u56f4\u662f[4,9]
a=(sinx)^2-sinx
有解
所以我们只需要求出(sinx)^2-sinx的取值范围即可
设g(x)=(sinx)^2-sinx=(sinx)^2-sinx+1/4-1/4=(sinx-1/2)^2-1/4
因为sinx∈[-1,1]
所以g(x)最大=(-1-1/2)^2-1/4=9/4-1/4=2
g(x)最小=-1/4
所以a∈[-1/4,2]
设sinx=t
-t^2+t+a=0
开口向下，有实数解
则德尔塔>=0
1+4a>=0
4a>=-1
a>=-1/4
f(x)=-(sinx)^2+sinx+a =-(sinx -1/2)^2 +1/4 +a
-1 <= sinx <= 1
-3/2 <= sinx -1/2 <=1/2
0 <= (sinx -1/2)^2 <=9/4
f(x)=0
0=-(sinx -1/2)^2 +1/4 +a
1/4 +a = (sinx -1/2)^2
0<= 1/4 +a <=9/4
-1/4 <= a <= 2
解：设sinx=y
y^2-t-a=0
有实数解，判别式△≥0
1+4a≥0
a>=-1/4
y=(1±√（1+4a))/2
注意到-1≤y≤1
故-1≤(1±√（1+4a))/2≤1
联合解得-1/4≤a≤0
解：方程f(x)=0就是：sin²x-sinx=a.===>[sinx-(1/2)]²=(4a+1)/4.∵-1≤sinx≤1.∴0≤[sinx-(1/2)]²≤9/4.即0≤(4a+1)/4≤9/4.===>-1/4≤a≤2.即a∈[-1/4,2].
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