Question

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

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

Answer 1

Answer: (D)

laadsc Consider a function $g(x)=xf(x)$
Obviously $g$ is continuous in $[0,1]$ and differentiable in $(0,1)$
As $f(1)=0$
$\therefore g(0)=0=g(1)$
Hence Rolle's theorem is applicable for $g$
$\therefore \exists$ some $c\in (0,1)$ such that
$g^{\prime }(c)=0$
$x{f}^{\prime }(x)+f(x)=0$
$c{f}^{\prime }(c)+f(c)=0$