Question

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

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

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

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

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

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