If f(x) is periodic function with period, T, then

Options

A. f and f' are also periodic
B. f is periodic but f' is not periodic
C. f is periodic but f' is not periodic
D. none of these

Answer

A

Explanation

Let f(x) be a periodic function with period. Then, `f(x+T)=f(x)` for all x
Now, `f'(x)=underset(h to 0)lim (f(x+h)-f(x))/(h)`
`therefore f'(x+T)=underset(h to 0)lim (f(x+T+h)-f(x+T))/(h)`
`Rightarrow f'(x+T)=underset(h to 0)lim (f(x+h)-f(x))/(h)=f'(x)"for all x"`
Similarly, we have
`f'(x+T)=f'(x)"for all x"`
Hence, f' and f'' are also periodic functions with the same period.