设f(x)在[0,a]连续,在(0,a)内可导,且f(a)=0,证明;存在一点c属于(0,a),使c^2f(c)+2cf(c)=0是c^2f(c)+2cf'(c)=0,我确定希望那位大师帮帮忙

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设f(x)在[0,a]连续,在(0,a)内可导,且f(a)=0,证明;存在一点c属于(0,a),使c^2f(c)+2cf(c)=0是c^2f(c)+2cf'(c)=0,我确定希望那位大师帮帮忙
设f(x)在[0,a]连续,在(0,a)内可导,且f(a)=0,证明;存在一点c属于(0,a),使c^2f(c)+2cf(c)=0
是c^2f(c)+2cf'(c)=0,我确定希望那位大师帮帮忙

设f(x)在[0,a]连续,在(0,a)内可导,且f(a)=0,证明;存在一点c属于(0,a),使c^2f(c)+2cf(c)=0是c^2f(c)+2cf'(c)=0,我确定希望那位大师帮帮忙
应该是c²f'(c)+2cf(c) = 0吧.
设g(x) = x²f(x),则g(c)在[0,a]连续,在(0,a)可导,且g(0) = 0 = g(a).
由罗尔定理,存在c∈(0,a)使g'(c) = 0,即有c²f'(c)+2cf(c) = 0.
不可能是c²f(c)+2cf'(c) = 0.
反例如f(x) = (x-a)·e^(-x²/4),易验证满足题目条件.
f'(x) = -x/2·e^(-x²/4)·(x-a)+e^(-x²/4) = (-x²+ax+2)/2·e^(-x²/4).
2xf'(x) = (-x³+ax²+2x)·e^(-x²/4).
x²f(x) = (x³-ax²)·e^(-x²/4).
则x²f(x)+2xf'(x) = 2x·e^(-x²/4) > 0对任意x > 0.
即不存在c > 0使c²f(c)+2cf'(c) = 0.

好像忘记了

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