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

Answer: (B)

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

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