函数f(x)对于任意非负实数x,y都满足f(x+y^2)=f(x)+2[f(y)]^2,且f(x)》0,f(x)不等于0,则f(2+根号3）=?

函数f(x)对于任意非负实数x,y都满足f(x+y^2)=f(x)+2[f(y)]^2,且f(x)》0,f(x)不等于0,则f(2+根号3）=?
函数f(x)对于任意非负实数x,y都满足f(x+y^2)=f(x)+2[f(y)]^2,且f(x)》0,f(x)不等于0,则f(2+根号3）=?

函数f(x)对于任意非负实数x,y都满足f(x+y^2)=f(x)+2[f(y)]^2,且f(x)》0,f(x)不等于0,则f(2+根号3）=?
f(2＋√3)＝f(√3＋(√2))²＝f(√3)＋2[f(√2)]²
f(0)＝f(0)＋2f²(0) ∴f(0)＝0
∴f(1)＝f(0＋1²)＝f(0)＋2f²(1) 2f²(1)＝f(1)
x＞0,f(x)≠0 ∴f(1)＝1/2
f(2)＝f(1)＋2f²(1) ＝1 f(3)＝f(2)＋2f²(1)＝3/2
2f²(√2)＋f(0)＝f(2)＝1 2f²(√2)＝1
又x＞0,f(x)≠0 ∴f(√2)＝√2/2
2f²(√3)＋f(0)＝f(3)＝3/2 2f²(√3)＝3/2 x＞0,f(x)≠0 f(√3)＝√3/2
所以 f(2＋√3)＝f(√3＋(√2))²＝f(√3)＋2[f(√2)]²＝√3/2＋2×1/2＝1＋√3/2

f(2＋√3)＝f(√3＋(√2))²＝f(√3)＋2[f(√2)]²
f(0)＝f(0)＋2f²(0) ∴f(0)＝0
∴f(1)＝f(0＋1²)＝f(0)＋2f²(1) ∴2f²(1)＝f(1) ∵x＞0,f(x)≠0 ∴f(1)＝1/2
∴f(2)＝f(1)＋...

全部展开

f(2＋√3)＝f(√3＋(√2))²＝f(√3)＋2[f(√2)]²
f(0)＝f(0)＋2f²(0) ∴f(0)＝0
∴f(1)＝f(0＋1²)＝f(0)＋2f²(1) ∴2f²(1)＝f(1) ∵x＞0,f(x)≠0 ∴f(1)＝1/2
∴f(2)＝f(1)＋2f²(1) ＝1 ∴f(3)＝f(2)＋2f²(1)＝3/2
∴2f²(√2)＋f(0)＝f(2)＝1 ∴2f²(√2)＝1 ∵x＞0,f(x)≠0 ∴f(√2)＝√2/2
∴2f²(√3)＋f(0)＝f(3)＝3/2 ∴2f²(√3)＝3/2 ∵x＞0,f(x)≠0 ∴f(√3)＝√3/2
∴f(2＋√3)＝f(√3＋(√2))²＝f(√3)＋2[f(√2)]²＝√3/2＋2×1/2＝1＋√3/2

收起