**Question :**

Let $$f\ :\ [a,b]\rightarrow \mathbb R$$ be continuous . And there exist constants $\alpha$ $\beta$ such that $\forall c\in [a,b]$ $$\alpha \int\_a^c f(x)dx +\beta \int\_c^b f(x)dx =0 $$ 

Prove that $f=0$ allover $[a,b]$.

**Attempt :**(*not* *much* *useful* ) 

Since the equation holds for all $c\in [a,b]$ , if take $c=a$ then we have $$\beta \int\_a^b f(x)dx =0 $$ 
And taking $c=b$ we get $$\alpha \int\_a^b f(x)dx=0 $$

See if both $\alpha=0$ and $\beta=0$ hold then there is nothing to do . If only one of them is $0$ and the other is non-zero then from the last two equations we get $$\int\_a^b f(x)dx =0 $$ and this does not ensure $f$'s being identically $0$ on $[a,b]$ (for example : $f(x)=sin x$ and $a=0$,$b=2\pi $ ) . 

So our only possibility is that both $\alpha \neq 0$ and $\beta \neq 0$ must hold .

Now I cannot proceed any further from this point . Please give me some hints as to what to do next .

Thaanks.