Let f, g be functions continuous on $[\mathrm{a},\mathrm{b}]$ and differentiable in (a,b) and let T: true and F: False consider the following statements

(A) If $\mathrm{f}\left(\mathrm{a}\right)=\mathrm{f}\left(\mathrm{b}\right)=0$ then for  such that $\mathrm{\beta }\mathbf{f}\left(\mathrm{\alpha }\right)+{\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{\alpha }\right)=0$ 

(B) If $\mathrm{f}\left(\mathrm{a}\right)=\mathrm{f}\left(\mathrm{b}\right)=0 \mathrm{\exists }\mathrm{\alpha }\in (\mathrm{a},\mathrm{b})$ such that ${\mathrm{g}}^{\mathrm{\prime }}\left(\mathrm{\alpha }\right)\mathrm{f}\left(\mathrm{\alpha }\right)+{\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{\alpha }\right)=0$ 

(C) If f(x) and g(x) are never zero on [a, b] and f(a)g(b) = f(b)g(a) then $\exists \alpha \in (\text{a},\text{b)}$ such that $\frac{{\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{\alpha }\right)}{\mathrm{f}\left(\mathrm{\alpha }\right)}=\frac{{\mathrm{g}}^{\mathrm{\prime }}\left(\mathrm{\alpha }\right)}{\mathrm{g}\left(\mathrm{\alpha }\right)}$  

(D) If $\left(\mathrm{f}\right(\mathrm{b}){)}^2-(\mathrm{f}\left(\mathrm{a}\right){)}^2={\mathrm{b}}^2-{\mathrm{a}}^2$ then ${\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{x}\right)\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}$ has no root in (a, b)

Identify the correct statements (where T and F are mentioned in order for the above statements)