Calculus~

MaoMao92

Let function f define as f(x+y)=f(x)+f(y) for all real values of x and y.
Prove that (a)f(0)=0
                   (b) f is an odd function
                   (c) If f'(0)=2 find the value of f'(3)

[ 本帖最后由 MaoMao92 于 22-3-2009 07:13 PM 编辑 ]

笨蛋一个

1) let x=0
f(x+y)=f(x)+f(y)
f(y)=f(0)+f(y)
f(0)=0

2)let x+y=0
x=-y
f(x+y)=f(x)+f(y)
f(0)=f(x)+f(-x)
f(x)=-f(-x)
f is an odd function

3)f(x+y)=f(x)+f(y)
f'(x+y)(1+dy/dx)=f'(x)+f'(y)(dy/dx)
let x+y=0 and x=3, y=-3
f'(0)(1+dy/dx)=f'(3)+f'(-3)(dy/dx)
but f(x)=-f(-x)
f'(x)=f'(-x) and f'(0)=2
2(1+dy/dx)=f'(3)+f'(3)(dy/dx)
f'(3)=2

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