F(x+y)=f(x)+f(y)+2xy f'(0)=2fx定义域为r,求fx
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F(x+y)=f(x)+f(y)+2xyf''(0)=2fx定义域为r,求fxF(x+y)=f(x)+f(y)+2xyf''(0)=2fx定义域为r,求fxF(x+y)=f(x)+f(y)+2xyf''(0
F(x+y)=f(x)+f(y)+2xy f'(0)=2fx定义域为r,求fx
F(x+y)=f(x)+f(y)+2xy f'(0)=2
fx定义域为r,求fx
F(x+y)=f(x)+f(y)+2xy f'(0)=2fx定义域为r,求fx
假设y > 0,记y = Δx,则有
f(x + Δx) - f(x) = f(Δx) + 2xΔx
从而有
[f(x + Δx) - f(x)]/Δx = f(Δx) + 2x
对上式取极限,使得Δx趋近于0+,则左式就是f'(x),即
f'(x) = limf(Δx) + lim2x =f(0) + 2x
所以现在要把f(0)搞定,令x = 0,有
f'(0) = f(0) = 2
所以
f'(x) = 2x + 2
f(x) = x^2 + 2x + C
代入f(0) = 2,有C = 2
所以f(x) = x^2 + 2x + 2
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