let $f(x)$ is continuous on $[a,b]$,and such $$\int_{0}^{1}x^kf(x)dx=0,k=0,1,2,3,\cdots,n$$
show that: there exsit $n+1$ different $a_{1},a_{2},\cdots,a_{n},a_{n+1}(a_{i}\neq a_{j},\forall i,j\in[1,n+1])\in [a,b]$,
such that $$f(a_{1})=f(a_{2})=\cdots =f(a_{n})=f(a_{n+1})=0$$
It is easy to prove $n=1$, Let $$F(x)=\int_{0}^{x}f(t)dt,F(0)=F(1)=0$$ and $$\int_{0}^{1}xf(x)dx=\int_{0}^{1}xdF(x)=xF(x)|_{0}^{1}-\int_{0}^{1}F(x)dx=0$$ so $$\int_{0}^{1}F(x)dx=0$$ so there exsit $\xi\in(0,1)$ such $F(\xi)=0$ so $$F(0)=F(1)=F(\xi)=0$$ so $$F'(\eta_{1})=F'(\eta_{2})=0$$ so $$f(\eta_{1})=f(\eta_{2})=0,\eta_{1}\neq\eta_{2}$$ so it prove by done,But for general,I can't
because I can't comment, Paramanand Singh,Hello,I read your solution, and you $d=e$ is possible,so I think your methods have some problem
