$Let \; f(x) = \left\{\begin{array}{ll} 1 + sinx \; , x < 0 \\ x^2 - x + 1 \; , x \geq 0 \end{array} \right.$ then

Q: $\large Let \; f(x) = \left\{\begin{array}{ll} 1 + sinx \; , x < 0 \\ x^2 – x + 1 \; , x \geq 0 \end{array} \right.$ then

(A) f has a local maximum at x = 0

(B) f has a local minimum at x=0

(C) f is increasing every where

(D) f is decreasing everywhere

Sol. f is continuous at ‘ 0 ’ and f'(0) > 0 and f’ ( 0+) < 0.

Thus f has a local maximum at ‘ 0 ’

Hence (A) is correct.