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

Answer 1

Answer: (A), (C)

A unique point in the interval $(n+\frac{1}{2},n+1)$

$f\left(x\right)=x\sin \pi x$
$f\left(x\right)=\sin \pi x+\pi x\cos \pi x=0$
$\Rightarrow -\tan \pi x=\pi x$

Clear $f^{\prime }(x$) has one root in $(n+\frac{1}{2},n+1)$, also $f^{\prime }\left(x\right)$ has one root in $(n,n+1)$.