定义在R上的单调函数f(x)满足f(x+y)=f(x)+f(y),且f(1)=2(1)求f(0)的值 (2)求证f(-x)=-f(x) (3)若f(kx)+f(x-x^2-2)<0对一切x∈R恒成立,求实数k的取值范围
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![定义在R上的单调函数f(x)满足f(x+y)=f(x)+f(y),且f(1)=2(1)求f(0)的值 (2)求证f(-x)=-f(x) (3)若f(kx)+f(x-x^2-2)<0对一切x∈R恒成立,求实数k的取值范围](/uploads/image/z/7303502-38-2.jpg?t=%E5%AE%9A%E4%B9%89%E5%9C%A8R%E4%B8%8A%E7%9A%84%E5%8D%95%E8%B0%83%E5%87%BD%E6%95%B0f%28x%29%E6%BB%A1%E8%B6%B3f%28x%2By%29%3Df%28x%29%2Bf%28y%29%2C%E4%B8%94f%281%29%3D2%EF%BC%881%EF%BC%89%E6%B1%82f%280%29%E7%9A%84%E5%80%BC+%EF%BC%882%EF%BC%89%E6%B1%82%E8%AF%81f%28-x%29%3D-f%28x%29+%EF%BC%883%29%E8%8B%A5f%28kx%29%2Bf%28x-x%5E2-2%29%EF%BC%9C0%E5%AF%B9%E4%B8%80%E5%88%87x%E2%88%88R%E6%81%92%E6%88%90%E7%AB%8B%2C%E6%B1%82%E5%AE%9E%E6%95%B0k%E7%9A%84%E5%8F%96%E5%80%BC%E8%8C%83%E5%9B%B4)
定义在R上的单调函数f(x)满足f(x+y)=f(x)+f(y),且f(1)=2(1)求f(0)的值 (2)求证f(-x)=-f(x) (3)若f(kx)+f(x-x^2-2)<0对一切x∈R恒成立,求实数k的取值范围
定义在R上的单调函数f(x)满足f(x+y)=f(x)+f(y),且f(1)=2
(1)求f(0)的值
(2)求证f(-x)=-f(x)
(3)若f(kx)+f(x-x^2-2)<0对一切x∈R恒成立,求实数k的取值范围
定义在R上的单调函数f(x)满足f(x+y)=f(x)+f(y),且f(1)=2(1)求f(0)的值 (2)求证f(-x)=-f(x) (3)若f(kx)+f(x-x^2-2)<0对一切x∈R恒成立,求实数k的取值范围
(1)f(x+y)=f(x)+f(y),令x=y=0
f(0+0)=f(0)+f(0)
f(0)=0
(2)令x=x,y=-x
f(x-x)=f(x)+f(-x)
f(0)=f(x)+f(-x)
f(x)=-f(-x)
(3)由(2)知,函数f(x)为奇函数,又因为f(0)=0,f(1)=2>0,可知当x>0,f(x)>0当x
(2)令x=x,y=-x
f(x-x)=f(x)+f(-x)
f(0)=f(x)+f(-x)
f(x)=-f(-x)
(3)由(2)知,函数f(x)为奇函数,又因为f(0)=0,f(1)=2>0,可知当x>0,f(x)>0当x<0,f(x)<0
f(kx)+f(x-x^2-2)=f(kx+x-x^2-2)<0
所以kx+x-x^2-2<0
整...
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(2)令x=x,y=-x
f(x-x)=f(x)+f(-x)
f(0)=f(x)+f(-x)
f(x)=-f(-x)
(3)由(2)知,函数f(x)为奇函数,又因为f(0)=0,f(1)=2>0,可知当x>0,f(x)>0当x<0,f(x)<0
f(kx)+f(x-x^2-2)=f(kx+x-x^2-2)<0
所以kx+x-x^2-2<0
整理得x^2-(k+1)x+2>0
可知方程对应的抛物线开口向上,因此若要x∈R恒成立,即△<0
△=b^2-4ac=(1+k)^2-8<0
即(1+k)^2<8
解得-1-2√2
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