If `f(x) = {{:(|x|",", "for",,0 lt |x| le 2), (1",", "for,, x =0):}`. Then, at x = 0, ` f ` has

Options

A. f(x) is decreasing on (-1,1) and has local minimum at x=1
B. f(x) is increasing on (-1,1) and has local minimum at x=1
C. f(x) is increasing on(-1,1) and has neither a local maximum nor a local minimum at x=1
D. f(x) is crdeeasing on(-1,1) but has neither a local maximum nor a local minimum at x=1

Answer

A

Explanation

We have
`f'(x)=(x^2-ax+1)/(x^2+ax+1),0lt alt2`
`rArr f'(x)=((x^2+ax+1)(2x-a)-(x^2-ax+1)(2x+a))/((x^2+ax+1)^2)`
`rArr f'(x) =(2a(x^2-1))/((x^2+ax+1)^2)`
The signs of f'(x) for different value of x are as under
So, f(x) is decreasing on (-1,1) and has a local maximum at x=-1 and local minimum at x=1