设 $fin C^2[0,1]$, $$ex f(0)=-1,quad f'(1)=3,quad int_0^1 xf''(x) d x=1. eex$$ 试求 $f(1)$.
解答: $$eex ea 1&=int_0^1 x d f'(x)\ &=xf'(x)|_0^1-int_0^1 f'(x) d x\ &=f'(1)-[f(1)-f(0)]\ &=f'(1)+f(0)-f(1)\ &=3-1-f(1) eea eeex$$ 知 $f(1)=1$.
设 $fin C^2[0,1]$, $$ex f(0)=-1,quad f'(1)=3,quad int_0^1 xf''(x) d x=1. eex$$ 试求 $f(1)$.
解答: $$eex ea 1&=int_0^1 x d f'(x)\ &=xf'(x)|_0^1-int_0^1 f'(x) d x\ &=f'(1)-[f(1)-f(0)]\ &=f'(1)+f(0)-f(1)\ &=3-1-f(1) eea eeex$$ 知 $f(1)=1$.