$Let \; f(x) = \left\{\begin{array}{ll} |x-1| + a \; , x < 1 \\ 2x + 3 \; , x \geq 1 \end{array} \right.$

Q: $\large Let \; f(x) = \left\{\begin{array}{ll} |x-1| + a \; , x < 1 \\ 2x + 3 \; , x \geq 1 \end{array} \right.$ If f(x) has a local minima at x = 1 , then

(A) a ≥ 5

(B) a > 5

(C) a > 0

(D) none of these

Sol. f(x) = 1- x + a , x < 1

= 2x + 3 , x  ≥ 1

Local minimum value of f(x) at x = 1, will be 5

i.e. 1- x + a  ≥ 5 at x = 1 or, a  ≥ 5.

Hence (A) is correct.