A firm’s profit function is differentiable for all production levels and its…

MATHEMATICS · Single correct · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

A firm’s profit function

P

is differentiable for all production levels

q>0

and its derivative is given by

P'(q)=60-1.5q

. Which of the following best justifies that producing 40 units yields a maximum profit for the firm? (a) Because

P'(40)=0

and

P''(40)>0

, indicating the profit curve is concave up at

q=40

. (b) Because

P'(40)=0

and

P''(40)<0

, indicating the profit curve is concave down at

q=40

. (c) Because

P'(40)=0

and

P'(q)<0

for all

q>40

, so profit decreases once production exceeds 40 units. (d) Because

P'(q)

is zero only at

q=40

, making 40 units the absolute maximum by uniqueness of the critical point.