Let the demand relation for a product be with

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Let the demand relation for a product be

p = a - bx

with

a,b>0

, and let the total cost of producing

x

units be

C(x)=c+dx+ex^{2}, \quad c,d,e>0.

Answer the following questions symbolically in terms of the parameters

a,b,c,d,e

. (i) Derive the profit function

P(x)

. (ii) Obtain an explicit formula for the output level

x^{*}

that maximises the profit. (iii) Show that the second derivative of

P(x)

evaluated at

x^{*}

is negative provided

e>b

, and interpret this result. (iv) Determine an inequality involving

a,b,c,d,e

that guarantees the maximum profit found in part (ii) is positive.