Discussion Forum

Let  f(x)= x sin  $\pi$ x, x>0 . Then for all natural numbers n, f'(x) vanishes at 

Answer:

Explanation:

Concept involved

 This question is based on the concept of transcendal  equation and sholud be solved with the help of graph

f(x)= x sin  $\pi$ x, x>0

$f'(x)= sin\pi x+\pi xcos \pi x=0$

$\Rightarrow$   $ tan \pi x=-\pi x$

 let f(x) = tan $\pi$ x  and g(x)= - $\pi$ x

 which could be  plotted as,

 clearly , f(x) and g(x)  intersect when

 $\frac{1}{2}< x<1$

  or            $\frac{3}{2}< x<2$

 or     $\frac{5}{2}< x<3$

 $\therefore$  a unique point in   $(n+\frac{1}{2},n+1)$

 or  (n,n+1)