已知f(x+y)=f(x)+f(y)+xY(x+y),f’(0)=1.求f(x).
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已知f(x+y)=f(x)+f(y)+xY(x+y),f’(0)=1.求f(x).
![已知f(x+y)=f(x)+f(y)+xY(x+y),f’(0)=1.求f(x).](/uploads/image/z/18472811-59-1.jpg?t=%E5%B7%B2%E7%9F%A5f%28x%2By%29%3Df%28x%29%2Bf%28y%29%2BxY%28x%2By%29%2Cf%E2%80%99%280%29%3D1.%E6%B1%82f%28x%29.)
f(x+y)=f(x)+f(y)+xy(x+y),①
令x=y=0得f(0)=2f(0),f(0)=0.
令y=-x,得f(0)=f(x)+f(-x),
∴f(x)是奇函数.
令y=x,得f(2x)=2f(x)+2x^3,
猜f(x)=ax^3+bx,
8ax^3+2bx=(2a+2)x^3+2bx,
比较系数得8a=2a+2,a=1/3,
f(x)=(1/3)x^3+bx,
f'(x)=x^2+b,
f'(0)=b=1.
∴f(x)=(1/3)x^3+x.
检验:f(x+y)=(1/3)(x+y)^3+x+y
=(1/3)x^3+x+(1/3)y^3+y+xy(x+y)
=f(x)+f(y)+xy(x+y).
f1(x)=(1/3)x^3+x.
下面证明本题仅此一设f(x)=f1(x)+g(x),由①,
f1(x+y)+g(x+y)=f1(x)+g(x)+f1(y)+g(y)+xy(x+y),
∴g(x+y)=g(x)+g(y),
∴g(x)=kx(k为常数),f(x)=f1(x)+kx,
f'(x)=f1'(x)+k,
令x=0得1=1+k,k=0,
∴f(x)=f1(x).
令x=y=0得f(0)=2f(0),f(0)=0.
令y=-x,得f(0)=f(x)+f(-x),
∴f(x)是奇函数.
令y=x,得f(2x)=2f(x)+2x^3,
猜f(x)=ax^3+bx,
8ax^3+2bx=(2a+2)x^3+2bx,
比较系数得8a=2a+2,a=1/3,
f(x)=(1/3)x^3+bx,
f'(x)=x^2+b,
f'(0)=b=1.
∴f(x)=(1/3)x^3+x.
检验:f(x+y)=(1/3)(x+y)^3+x+y
=(1/3)x^3+x+(1/3)y^3+y+xy(x+y)
=f(x)+f(y)+xy(x+y).
f1(x)=(1/3)x^3+x.
下面证明本题仅此一设f(x)=f1(x)+g(x),由①,
f1(x+y)+g(x+y)=f1(x)+g(x)+f1(y)+g(y)+xy(x+y),
∴g(x+y)=g(x)+g(y),
∴g(x)=kx(k为常数),f(x)=f1(x)+kx,
f'(x)=f1'(x)+k,
令x=0得1=1+k,k=0,
∴f(x)=f1(x).
f(xy)=f(x)+f(y),证明f(x/y)=f(x)-f(y)
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