Mathematics

Problem

(Long Answer – Contextual Application) A small workshop produces a handcrafted item whose production cost (in rupees) for

x

units is given by

C(x)=\begin{cases} 2000+60x, & 0\le x\le 50,\\ 5000+40|x-50|, & x>50. \end{cases}

Assume

x

is a whole number and

x\le 100

. (a) Show that

x=50

is a critical point of

C

. (b) Determine the nature of this critical point by examining the sign of the appropriate difference of costs for

x

just below and just above

50

. (c) State the production level(s) at which the workshop incurs minimum cost within

0\le x\le 100

and write down that minimum cost.