Question

Let \(f(x)=\cos \left(\frac{\pi}{x}\right), x \neq 0\) then assuming \(k\) as an integer,
(A) \(f(x)\) increases in the interval \(\left(\frac{1}{2 k+1}, \frac{1}{2 k}\right)\)
(B) \(f(x)\) decreases in the interval \(\left(\frac{1}{2 k+1}, \frac{1}{2 k}\right)\)
(C) \(f(x)\) decreases in the interval \(\left(\frac{1}{2 k+2}, \frac{1}{2 k+1}\right)\)
(D) \(f(x)\) increases in the interval \(\left(\frac{1}{2 k+2}, \frac{1}{2 k+1}\right)\)

Answer 1

Ans: (A, C)
Hint: \(f^{\prime}(x)=\sin \left(\frac{\pi}{x}\right)\left(\frac{\pi}{x^{2}}\right)\)
$$
\begin{array}{ll}
f^{\prime}(x)>0 & x \in\left(\frac{1}{2 k+1}, \frac{1}{2 k}\right) \\
f^{\prime}(x)<0 & x \in\left(\frac{1}{2 k+2}, \frac{1}{2 k+1}\right)
\end{array}
$$