A manufacturer models the profit (in thousand rupees) from producing and selling…

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

Problem

A manufacturer models the profit (in thousand rupees) from producing and selling

x

units of a product by

P(x)=-x^{3}+3b x^{2}+12x-5,

where

b

is a real constant that can be adjusted by changing the marketing budget. Management requires that the maximum profit occur when

x=4

. (i) Determine the value of

b

that satisfies the management requirement. (ii) Using this value of

b

, compute the maximum profit. (iii) For the same value of

b

, examine whether any other stationary point exists and classify its nature.