# 3.x articles are produced at a total cost of (2x^2+30x+20)naira and each is sold for (x/3+100) naira. Find the of x which gives the greatest profit?

Aug 12, 2018

3.x articles are produced at a total cost of $\left(2 {x}^{2} + 30 x + 20\right)$ naira
and each is sold for $\left(\frac{x}{3} + 100\right)$ naira.

So sale price $= 3 x \left(\frac{x}{3} + 100\right) = 3 {x}^{2} + 300 x$

Hence profit $P = 2 {x}^{2} + 30 x + 20 - 3 {x}^{2} - 300 x$

$\implies P = 20 - {x}^{2} - 270 x$

=>P=20-(x^2-2*x*135+135^2-135^2

$\implies P = 20 + 18225 - {\left(x - 135\right)}^{2}$

So prfit will be maximua\m when $x = 135$