Let $f$ , $g$ be real continuous functions in $[a,b]$. Prove that there is $c\in(a,b)$ such that
What would you suggest me to do here? Thanks.
Let us consider the function $$ H(x)=(b-x)\int_a^xf+(x-a)\int_x^bg $$ This is $C^1$ and $H(a)=H(b)=0$.
So by Rolle, there exists $c\in (a,b)$ such that $H'(c)=0$.