Question

Let $f\left(x\right)=x\sin \pi x,x>0$. Then for all natural numbers $n,f^{\prime }\left(x\right)$ vanishes at :

• A. a unique point in the interval $(n,n+\frac{1}{2})$
• B. a unique point in the interval $(n+\frac{1}{2},n+1)$
• C. a unique point in the interval $(n,n+1)$
• D. two points in the interval $(n,n+1)$

Answer 1

Answer: (B), (C)

The correct options are
B a unique point in the interval $(n+\frac{1}{2},n+1)$
C a unique point in the interval $(n,n+1)$

Given that $f\left(x\right)=x\sin \pi x$

$\Rightarrow f^{\prime }\left(x\right)=\sin \pi x+\pi x\cos \pi x$

$f^{\prime }\left(x\right)=0$

$\Rightarrow \sin \pi x+\pi x\cos \pi x=0$

$\Rightarrow \tan \pi x=-\pi x$

$\Rightarrow \pi x\in (\frac{2n+1}{2}\pi ,(n+1\left)\pi \right)\Rightarrow x\in (n+\frac{1}{2},n+1)$ and also $x\in (n,n+1)$.