$(7$ marks$)$ Let $f(x)$ and $g(x)$ be two differentiable functions on $0,1,$ with $f(0)=2,g(0)=0,f(1)=6,g(1)=2$ and $g^{\text{'}}(x)0$ for every xin$(0,1).$ Prove that there exists cin$(0,1)$ such that $f^{\text{'}}(c)=2g^{\text{'}}(c).$