Question

If $f\left(x\right)=\left[x\right]tan\left(\pi x\right)$ then $f^{\prime }\left(k^{+}\right)$ is equal to (where k is some integer and $[.]$ denotes greatest integer function)

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

Answer 1

Answer: (B)

$k\pi$

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

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