suppose that f is a continuously differentiable function and
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suppose that f is a continuously differentiable function and f(x/3)=f(x)/3 for all x in R
a) show that f(0)=0
b) show that f ' is constant and in fact for all x,f ' (x)=f(1)
a) show that f(0)=0
b) show that f ' is constant and in fact for all x,f ' (x)=f(1)
a) 取x=0即可
b) 求导得f'(x)=f'(x/3),所以f'(x)=f'(x/3)=...=f'(x/3^n),由导函数的连续性得f'(x)=f'(0)
再由中值定理f(1)=f(1)-f(0)=f'(xi)=f(x)
再问: thanks!
b) 求导得f'(x)=f'(x/3),所以f'(x)=f'(x/3)=...=f'(x/3^n),由导函数的连续性得f'(x)=f'(0)
再由中值定理f(1)=f(1)-f(0)=f'(xi)=f(x)
再问: thanks!
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