MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
For each real number k, define pk(x)=ekx−kx for all real x.
(a) Obtain a simplified expression for pk′(x) in terms of k and x.
(b) Determine all real values of k for which pk is strictly increasing on the entire real line. Provide complete justification.
(c) Determine all real values of k for which pk has exactly one critical number. Show all reasoning, including any necessary discussion of the existence or non–existence of additional critical numbers.
(d) For each k found in part (c), decide whether the corresponding critical point yields a relative maximum, a relative minimum, or neither for pk. Justify your answer fully in each case.