Application of Mean Value Theorem on 2 differentiable functions simultaneously

Question

MCQ question

Let $f$ and $g$ be two differentiable functions on $(0,1)$ such that $f(0)=2$, $f(1)=6$, $g(0)=0$ and $g(1)=2$

Then there exists $\theta \in (0,1)$ such that $f'(\theta)$ equals

(a) $1/2 g'(\theta)$

(b)$2g'(\theta)$

(c) $6g'(\theta)$

(d) $1/6 g'(\theta)$

Now I know that we have to apply Mean Value Theorem here. But my question is wouldn't we get two different $\theta$ that is $\exists\, \theta_{1}$ such that $f'(\theta_{1})=4$ and $\exists\, \theta_{2}$ such that $g'(\theta_{2})=2$

So, how can option (b) be the right option unless $\theta_{1}=\theta_{2}$?

Answer 1

Hint

Apply Rolle's theorem to the function $f-2g$.