Question

If $f$ be a continuous function on $[0,1]$, differentiable in $(0,1)$ such that $f\left(1\right)=0$, then there exists some $c\in (0,1)$ such that

• A. $c{f}^{\prime }\left(c\right)-f\left(c\right)=0$
• B. $f^{\prime }\left(c\right)+cf\left(c\right)=0$
• C. $f^{\prime }\left(c\right)-cf\left(c\right)=0$
• D. $c{f}^{\prime }\left(c\right)+f\left(c\right)=0$

Answer 1

The correct option is A $c{f}^{\prime }\left(c\right)-f\left(c\right)=0$

Now, $f^{\prime }\left(c\right)=\frac{f\left(c\right)-0}{c-0}$ (Formula of slope, when $2$ points are known)

$c{f}^{\prime }\left(c\right)-f\left(c\right)=0$ Answer.