已知二次函数f(x)=ax^2+bx满足f(1+x)=f(1-x)且方程f(x)=x有两个相等实数,若函数f(x)在定义域为[m,n]上
对应的值域为[2m,2n],求m,n的值
人气:325 ℃ 时间:2019-08-19 15:49:19
解答
f(x)=ax^2+bx=a(x+b/2a)^2-b^2/4a,则b/2a=-1,b=-2a;
f(x)=x=ax^2+bx,ax^2+bx-x=(ax+b-1)x=0,x=0或x=(1-b)/a两根相等,a≠0,故1-b=0,b=1;
所以,a=-1/2,f(x)=-x^2/2+x=-(x-1)^2/2+1/2;
当m>1时,f(x)为递减函数,则有f(n)=2m,f(m)=2n,代入得:
-n^2/2+n=2m,-m^2/2+m=2n,两式相加:-(m^2+n^2)/2=m+n,不成立;
当n1,m
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