# Question 6a90d

May 22, 2017

If $60$ items per day are produced, the unit cost is minimum, i.e.,

$400. #### Explanation: $C \left(x\right) = {x}^{2} - 120 x + 4000.$Completing the square, we get, $C \left(x\right) = {x}^{2} - 2 \cdot 60 x + {60}^{2} - {60}^{2} + 4000 , i . e . ,$$C \left(x\right) = {\left(x - 60\right)}^{2} + 4000 - 3600 = {\left(x - 60\right)}^{2} + 400 ,$Since, $\forall x \in \mathbb{R} , {\left(x - 60\right)}^{2} \ge 0 , C \left(x\right) \ge 400 ,$which means that, The Minimum Unit Cost is $400.

Further, for this minimum unit cost, if $x$ items per day are produced, then,

$C \left(x\right) = 400 \Rightarrow {\left(x - 60\right)}^{2} + 400 = 400 \Rightarrow x = 60.$

Thus, if daily $60$ items are produced, the unit cost is minimum, i.e.,

\$400.#

Enjoy Maths.!