Question

The left hand derivative of $f\left(x\right)=\left[x\right]\sin \pi x$ at $x=k,k$ is an integer, 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 ${(-1)}^k(k-1)\pi$

Given $f\left(x\right)=\left[x\right]\sin \pi x$

If x is just less than $k,\left[x\right]=(k-1)$

Therefore $f\left(x\right)=(k-1)\sin \pi x$

LHD of $f\left(x\right)=\underset{x\to k}{lim}\frac{(k-1)\sin \pi x-k\sin \pi k}{x-k}$
$=\underset{x\to k}{lim}\frac{(k-1)\sin \pi x}{x-k}=\underset{h\to 0}{lim}\frac{(k-1)\sin \pi (k-h)}{-h}$
Where $x=k-h$

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