Consider the piece-wise defined function f(x)= cases x^ 2 +ax, & x 1, 3x-a, & x>1
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
Consider the piece-wise defined function
f(x)={x2+ax,3x−a,x≤1,x>1,
where a is a real constant.
(i) Find the condition on a that makes f differentiable at x=1.
(ii) For the value(s) of a satisfying part (i), determine whether f has a local minimum at x=1.
(iii) For the same value(s) of a, identify the intervals on which f is increasing and those on which it is decreasing.