Question

The left-hand derivative of f(x) = [x] sin ($\pi$ x) at x = k, k is an integer and [x] = greatest integer $\le$ x, is

[IIT Screening 2001]

• A. $(-1{)}^k(k-1)\pi$
• B. $(-1{)}^{k-1}(k-1)\pi$
• C. $(-1{)}^{k}k\pi$
• D. $(-1{)}^{k-1}k\pi$

Answer 1

The correct option is A $(-1{)}^k(k-1)\pi$

f(k-0) = $\underset{k\to 0}{lim}\frac{[k-h]sin\pi (k-h)-\left[k\right]sin\pi k}{-h}$
= $\underset{k\to 0}{lim}\frac{(-1{)}^{k-1}(k-1)sin\pi h-k\times 0}{-h}$
= $\underset{k\to 0}{lim}\frac{(-1{)}^{k-1}(k-1)sin\pi h}{-h}=(-1{)}^k.(k-1)\pi .$