Question

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

• 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

Answer: (A)

$(-1{)}^k(k-1)\pi$

$LHD=\underset{h\to 0}{lim}f\left(x\right)=\underset{h\to 0}{lim}\frac{[k-h]\sin \pi (k-h)-\left[k\right]\sin k\pi }{-h}$

$=\underset{h\to 0}{lim}\frac{(k-1)\sin (k\pi -h\pi )-k\sin k\pi }{-h}$

$=\underset{h\to 0}{lim}\frac{\pm (k-1)\sin h\pi }{-h}$

$=\pm (k-1)\pi$

$=(-1{)}^k(k-1)\pi$