Question

If f and g are differentiable functions in $[0,1]$ satisfying $f\left(0\right)=2=g\left(1\right),g\left(0\right)=0$ and $f\left(1\right)=6$, then for some c $ϵ$ ]0,1[

• A. $f^{\prime }\left(c\right)=g^{\prime }\left(c\right)$
• B. $f^{\prime }\left(c\right)=2g^{\prime }\left(c\right)$
• C. $2f^{\prime }\left(c\right)=g^{\prime }\left(c\right)$
• D. $2f^{\prime }\left(c\right)=3g^{\prime }\left(c\right)$

Answer 1

The correct option is B $f^{\prime }\left(c\right)=2g^{\prime }\left(c\right)$

Since, f and g both are continuous functions on $[0,1]$ and differentiable on $(0,1)$ then $\exists cϵ(0,1)$ such that
$f^{\prime }\left(c\right)=\frac{f\left(1\right)-f\left(0\right)}{1}=\frac{6-2}{1}=4$
and $g^{\prime }\left(c\right)=\frac{g\left(1\right)-g\left(0\right)}{1}=\frac{2-0}{1}=2$
Thus, we get $f^{\prime }\left(c\right)=2g^{\prime }\left(c\right)$