高一 数学竞赛
f(x)对于任何非负数xy有f(x+y^2)=f(x)+2[f(y)]^2f(x)是正数,f(1)不为1,求f(2+根号3)...
f(x)对于任何非负数xy有f(x+y^2)=f(x)+2[f(y)]^2
f(x)是正数,f(1)不为1,求f(2+根号3) 展开
f(x)是正数,f(1)不为1,求f(2+根号3) 展开
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解：f(x+y^2)=f(x)+2[f(y)]^2
f(0)=f(0+0^2)=f(0)+2[f(0)]^2
f(0)=0
f(0+x^2)=f(0)+2[f(x]]^2
f(x)=2[f(√x)]^2
f(1)=f(0+1^2)=f(0)+2[f(1)]^2=0+2[f(1)]^2
f(1)=1/2
f(2)=f(1+1^1)=f(1)+2[f(1)]^2=1/2+1/2=1
f(3)=f(2+1)=f(2)+2[f(1)]^2=1+1/2=3/2=2[f(√3)]^2
f(√3)=√3/2
f(2+√3)=f(2)+2f[√(√3)]^2=1+f(√3)=1+√3/2
f(0)=f(0+0^2)=f(0)+2[f(0)]^2
f(0)=0
f(0+x^2)=f(0)+2[f(x]]^2
f(x)=2[f(√x)]^2
f(1)=f(0+1^2)=f(0)+2[f(1)]^2=0+2[f(1)]^2
f(1)=1/2
f(2)=f(1+1^1)=f(1)+2[f(1)]^2=1/2+1/2=1
f(3)=f(2+1)=f(2)+2[f(1)]^2=1+1/2=3/2=2[f(√3)]^2
f(√3)=√3/2
f(2+√3)=f(2)+2f[√(√3)]^2=1+f(√3)=1+√3/2
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