Question

A company determines that in order to sell x

x

 items the price per item, in dollars, must be p(x)=1830

p(x)=1830. The company also determines that the total cost, in dollars, to produce x x

 items is given by C(x)=3100+610x+1.5x2C(x)=3100+610x+1.5x2. 

How many items must the company produce and sell in order to maximize profit? 

The company must produce and sell 

Answer 1

$xp\left( x \right) -C\left( x \right) \rightarrow \max \\-\left( 3100+610x+1.5x^2 \right) +1830x\rightarrow \max \\-1.5x^2+1220x-3100\rightarrow \max \\x_{\max}=\frac{1220}{2\cdot 1.5}=406.667\approx 407\\The\,\,company\,\,should\,\,produce\,\,407 items$