A firm’s profit (in thousand rupees) from producing and selling hundred units of a…

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

Problem

A firm’s profit (in thousand rupees) from producing and selling

x

hundred units of a product is modelled by

P(x)=-x^{4}+4x^{3}+15,

where

0\le x\le 5

. Analyse this model as follows. (i) Find

P'(x)

and determine all stationary points of

P

in the given interval. (ii) Compute

P''(x)

and apply it to classify each stationary point, mentioning any point where

P''(x)=0

. (iii) For every point where the conclusion in part (ii) is inconclusive, investigate the sign of

P'(x)

on appropriate intervals to decide its nature. (iv) Using all the information above and the values of

P

at the endpoints, find the production level that gives the maximum profit and state the corresponding maximum profit.