Answer
Profit function : $P(x)=-100x-5000$
Break-even quantity : $x=-50$
Decision : Do not proceed with the production.
Work Step by Step
Given, the cost function as $C(x)=1000x+5000$ and revenue function as $R(x)=900x$. At the break even point, profit is zero. To find the break even quantity we equate :
$R(x)=C(x)$
$900x=1000x+5000 \implies x=-50$
We got a negative break even point which means that any amount of good you produce and sell will only result in a profit $<0$. Why is this so? Let's see.
The profit function is given by:
$P(x)=R(x)-C(x)$
$P(x)=-100x-5000 <0\hspace{0.1cm} \forall \hspace{0.1cm}x>0$
It is clear that the profit function is always negative for whatever quantity produced and sold. Hence, we shouldn't proceed with the production.
