A manufacturer’s cost

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

A manufacturer’s cost, in dollars, for producing

q

thousand units of a certain product is modeled by

C(q)=q^{3}-12q^{2}+60q+100

for

0\le q\le 15

. (a) Compute

C''(q)

. (b) Find every production level

q

in the domain where

C''(q)=0

C''(q)

is undefined. These are the only production levels at which the concavity of the cost function could change. (c) Determine on which subintervals of

[0,15]

the cost function is concave up and on which subintervals it is concave down. Provide clear reasoning based on your analysis of

C''(q)

. (d) Identify each point of inflection on the graph of

C

within the closed interval

[0,15]

, giving its coordinates

(q,C(q))

in exact form. (e) Interpret, in the context of production and cost, the economic meaning of a point of inflection on the graph of

C