[微分] 不等式

[t0127754 (阿彥)]

Let f be a real-valued differentiable function on R

s.t f'(x)>f(x) for all real number x.

Assume that f(0)=0 show that f(x)>0 for all x >0




--
※ 發信站: 批踢踢實業坊(ptt.cc) 
◆ From: 118.169.48.173
推 LeoRen:若f(x)>0 => f'(x)/f(x) >1 => lnf(x)>x
→ LeoRen:=> f(x)>e^x, f(0)>1 矛盾!
→ LeoRen:lnf(x)>x+c =>f(x)>exp(x+c),f(0)>0 矛盾
→ t0127754:所以題目錯了?
→ t0127754:不對阿 那是不定積分 要加常數C 才會對
→ t0127754:應該沒有矛盾吧
→ LeoRen:看我所舉的反例是否無誤,若是那題目就有誤
→ LeoRen:第3行便是
→ LeoRen:第2行是c=0的特例
推 Dazaiosamu:科
→ muxiv       : 所以題目錯了? https://moxox.com