Consider the piece-wise defined function
f(x)={x2+ax,3x−a,x≤1,x>1,
where a is a real constant.
a. (i) Find the condition on a that makes f differentiable at x=1.
b. (ii) For the value(s) of a satisfying part (i), determine whether f has a local minimum at x=1.
c. (iii) For the same value(s) of a, identify the intervals on which f is increasing and those on which it is decreasing.