1. 设 $fin C^2(bR)$, $f''(x)geq 0$, $f(0)=0$. 对 $0<a<b$, 试比较 $f(a+b)$ 与 $f(a)+f(b)$ 的大小.
解答: 设 $F(x)=f(x+a)-f(x)-f(a)$, 则 $F(0)=0$, $F'(x)=f'(x+a)-f'(x)=f''(xi_x)ageq 0$. 故 $F(b)>0$, $f(a+b)geq f(a)+f(b)$.
2. 设当 $x>-1$ 时, 可微函数 $f(x)$ 满足条件 $$ex f'(x)+f(x)-cfrac{1}{x+1}int_0^x f(t) d t=0, eex$$ 且 $f(0)=1$. 试求 $f'(x)$.
解答: $$eex ea &quad f'(x)+f(x)-cfrac{1}{x+1}int_0^x f(t) d t=0\ & a f'(x)+f(0)+int_0^x f'(s) d s -cfrac{1}{x+1}int_0^x sez{f(0)+int_0^t f'(s) d s} d t=0\ & a f'(x)+1+int_0^x f'(s) d s -cfrac{x}{x+1}-cfrac{1}{x+1}int_0^x (x-s)f'(s) d s=0\ & a f'(x)+cfrac{1}{x+1}+cfrac{1}{x+1}int_0^x (s+1)f'(s) d s=0\ & a (x+1)f'(x)+1+int_0^x(s+1)f'(s) d s=0\ & a F'(x)+1+F(x)=0quad sex{F(x)=int_0^x (s+1)f'(s) d s}\ & a [e^xF(x)]'=-e^x\ & a e^xF(x)=1-e^x\ & a F(x)=e^{-x}-1\ & a (x+1)f'(x)=-e^{-x}\ & a f'(x)=-cfrac{e^{-x}}{x+1}. eea eeex$$