Let f(x) is a polynomial function and `f(alpha))^(2) + f'(alpha))^(2) = 0`, then find `underset(x rarr alpha)(lim) (f(x))/(f'(x))[(f'(x))/(f(x))]`, where [.] denotes greatest integer function, is........

Options

A. 0
B. 1
C. -1
D. `f(alpha)f'(alpha)`

Answer

B

Explanation

It is given that the polynomial `f(x)` satisfies the relation `(f(alpha))^2+(f'(alpha))^2=0`
`therefore f(alpha)=0=f'(alpha)`
`rArr x=alpha` is a root of `f(x)` and `f'(x)`
`rArr (x-alpha)^2` is a foactor of `f(x)`.
Let `f(x)=(x-alpha)^2phi(x)`. Then,
`f'(x)=2(x-alpha)phi(x)+(x-alpha)^2phi'(x)`
`therefore (f(x))/(f'(x))=((x-alpha)phi(x))/(2phi(x)+(x-alpha)phi'(x))`
Now, `lim_(x to alpha) (f(x))/(f(x))[(f'(x))/(f(x))]`
` lim_( xtoalpha) (f'(x))/(f'(x))((f'(x))/(f'(x))-{f(x)/f(x)})`, whwre `[.]` denotes the fractional part function.
`=lim_(xtoalpha) (f(x))/(f(x))xx(f'(x))/(f(x))-lim_(xtoalpha)(f(x))/(f'(x)){(f'(x))/(f(x))}`
` =1-lim_(xtoalpha)(x-alpha)/(2phi(x)+(x-alpha)phi'(x))`
`{(2phi(x)+(x-alpha)phi'(x))/((x-alpha)phi(x))}`
`=10=1`.