A function f is defined piecewise by
f(x)={x2+4x+1,2x+k,x≤1,x>1,
where k is a real constant.
a. Part 1: (a) Determine the value of k that makes f continuous at x=1. Show algebraic work.
b. Part 2: (b) For the k found in part (a), compute the derivative of f on each interval and state any x-values where f fails to be differentiable.
c. Part 3: (c) List every critical number of f in its domain, including any point where f'(x)=0 or f is not differentiable. Justify why each point is critical.
d. Part 4: (d) Analyze the sign of f'(x) on each interval determined by the critical numbers. Hence classify every critical number as a local maximum, local minimum, or neither, and support your conclusions with clear reasoning.