Question

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

This equation is based on the concept of transcendal
equation and should be solved with the help of graph.
f (x) = x sin $\pi$ t, t > 0
f (x) = sin $\pi$ x + $\pi$ x cos $\pi$ 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) interest 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).