Let `f (x)= {{:(1+ sin x "," , x lt 0),( x ^(2) -x+1",", x ge 0):},` then:

Options

A. f has a local maximum at x=0
B. f has a local minimum at x=0
C. f is increasing in (0, 1/2)
D. f is decreasing in (0,1/2)

Answer

B

Explanation

we have
`underset(x to 0^-)limf(x)=underset(x to 0^-)lim1 +sin x=1`
`underset(xrarr0^-)limf(x)=underset(xrarr0^+)limx^2-x+1=1 and f(0)1`
So. F is continuous at x=0
Also (LHD at x =0 ) =`{d/dx (1+sin x)}_(at x =0)=- lt 0 `
and ,(RHD at x=0 )`={d/dx(x^2-x+1)}_(at x=0)=-1 lt 0 `
Thus, f(x) in increasing in the left neighbourhood of x=0 and decreasing in the right neighbourhood, So f(x) has a local maximum at x-0 ,
Also f(x)= cos ` x gt 0 for x in (-pi//2,0)`
and `f(x)=2x-1 lt 0 for x in (0,1//2)`
Thus f(x) is increasing in `(-pi//2, 0)` and decreasing in (0,1/2)