A function f is defined piecewise by f(x)= cases x^ 2 +4x+1, & x 1, 2x+k, & x>1
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A function f is defined piecewise by
f(x)={x2+4x+1,2x+k,x≤1,x>1,
where k is a real constant.
(1) Part 1: (a) Determine the value of k that makes f continuous at x=1. Show algebraic work.
(2) 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.
(3) 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.
(4) 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.