The left hand derivative of `f(x)=[x]sin(pix)` at `x = k`, `k` is an integer, is

Options

A. `(-1)^(k)(k-1)pi`
B. `(-1)^(k-1)(k-1)pi`
C. `(-1)^(k)kpi`
D. `(-1)^(k-1)kpi`

Answer

A

Explanation

`f(x)=[x]sin(pix)" at "x=k`.
Left hand derivative, `underset(hto0)lim(f(k)-f(k-h))/(h)` (k-integer)
`underset(hto0)lim(-(k-1)sin(k-h)pi)/(h)`
`sinkpi=0andsin(kpi-theta)=(-1)^(k-1)sintheta`.
`=underset(hto0)lim(-(k-1)-(1)^(k-1)sinpi)/(hpi)xxpi`
`=underset(hto0)lim(-(k-1)(-1)^(k-1)sinpi)/(hpi)xxpi`
`=pi(k-1)(-1)^(k)`