Suppose $a$ and $b$ from $\mathbb{R}$ as $a
How can I prove that $\exists$ $c$ $\in$ $]a;b[\;\;$ s.t. $\;\;\dfrac { f(b)-f(a) }{ g(b)-g(a) } =\dfrac { f^{ ' }\left( c \right) }{ g^{ ' }\left( c \right) } $ using Rolle's theorem.
Suppose $a$ and $b$ from $\mathbb{R}$ as $a
How can I prove that $\exists$ $c$ $\in$ $]a;b[\;\;$ s.t. $\;\;\dfrac { f(b)-f(a) }{ g(b)-g(a) } =\dfrac { f^{ ' }\left( c \right) }{ g^{ ' }\left( c \right) } $ using Rolle's theorem.
Hint
Define the function $$ h(x)=f(x)(g(b)-g(a))-g(x)(f(b)-f(a)) $$ then $h(b)=h(a)$. Apply Rolle's Theorem.
Look at the function $$h(x) = (f(b) - f(a))g(x) - (g(b) - g(a)) f(x)$$