Solving Functions Problems: Lagrange and Rolle's Theorem

[laura1231]

Hi, in a book I have found this problem:
"Let be $f,g:\mathbb{R}\rightarrow\mathbb{R}$ two derivable functions such that $f(0)=g(0)$ and $f(6)=g(6)$. Which of the following statements is necessarily true?:
a) $\exists\ c\in]0;6[ : f'(c)=g'(c)$;
b) $\exists\ c_1,c_2\in]0;6[ : f'(c_1)=g'(c_2)$.
"
The author of this question indicates the answer b because, for Lagrange's theorem $\exists\ c_1\in ]0;6[ : f'(c_1)=\dfrac{f(6)-f(0)}{6-0}$ and $\exists\ c_2\in ]0;6[ : g'(c_2)=\dfrac{g(6)-g(0)}{6-0}$, therefore $f'(c_1)=g'(c_2)$ but you can't be sure that $c_1=c_2$...
I think the author misses in fact if you call $h(x)=f(x)-g(x)$ then, the hypothesis of Rolle's theorem are true ($h(0)=h(6)$) therefore $\exists\ c \in ]0;6[: h'(c)=0$ then $f'(c)=g'(c)$. For me the correct answer is a.
Am I right?

 

[I like Serena]

Hi, in a book I have found this problem:
"Let be $f,g:\mathbb{R}\rightarrow\mathbb{R}$ two derivable functions such that $f(0)=g(0)$ and $f(6)=g(6)$. Which of the following statements is necessarily true?:
a) $\exists\ c\in]0;6[ : f'(c)=g'(c)$;
b) $\exists\ c_1,c_2\in]0;6[ : f'(c_1)=g'(c_2)$.
"
The author of this question indicates the answer b because, for Lagrange's theorem $\exists\ c_1\in ]0;6[ : f'(c_1)=\dfrac{f(6)-f(0)}{6-0}$ and $\exists\ c_2\in ]0;6[ : g'(c_2)=\dfrac{g(6)-g(0)}{6-0}$, therefore $f'(c_1)=g'(c_2)$ but you can't be sure that $c_1=c_2$...
I think the author misses in fact if you call $h(x)=f(x)-g(x)$ then, the hypothesis of Rolle's theorem are true ($h(0)=h(6)$) therefore $\exists\ c \in ]0;6[: h'(c)=0$ then $f'(c)=g'(c)$. For me the correct answer is a.
Am I right?

Hi laura! ;)

Yep. I believe you are right.
Therefore both statements are true.