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Q.
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
Application of Derivatives
Solution:
laadsc Consider a function $g ( x )= x f ( x )$
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$