Prove that $\int_a^cf(x)\mathrm{d}x+(c-a)g(c)=\int_c^bg(x)\mathrm{d}x+(b-c)f(c)$

Question

Let $f$ , $g$ be real continuous functions in $[a,b]$. Prove that there is $c\in(a,b)$ such that

$$\int_a^cf(x)\mathrm{d}x+(c-a)g(c)=\int_c^bg(x)\mathrm{d}x+(b-c)f(c)$$

What would you suggest me to do here? Thanks.

Answer 1

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 $H'(c)=0$.