设 $f(x)$ 在 $bR$ 上连续, 又 $$ex phi(x)=f(x)int_0^x f(t) d t eex$$ 单调递减. 证明: $fequiv 0$.
证明: 设 $$ex g(x)=cfrac{sez{int_0^x f(t) d t}^2}{2}, eex$$ 则 $g'(x)=phi(x)$ 递减, 而 $$ex g'(x)sedd{a{ll} geq g'(0)=0,&x<0,\ leq g'(0)=0,&x>0; ea} eex$$ 进一步, $$ex g(x)sedd{a{ll} leq g(0)=0,&x<0,\ leq g(0)=0,&x>0. ea} eex$$ 如此, $g(x)leq 0$, $$ex int_0^x f(t) d t=0,quad forall x, eex$$ $$ex f(x)=sez{int_0^x f(t) d t}'=0,quad forall x. eex$$