The cost of producing x items per day is given by y=12+3x+x^2. If each item can be sold for $10, determine the break-even point.

The cost of producing x items per day is given by y = 12 +
3x + x^2.

Each item can be sold for $10. The income made
when x items are sold is 10*x and the cost of production is 12 + 3x +
x^2.

The revenue function is the amount that one gets by
selling the items. This is f(x) = 10x

The profit function
is the total revenue - total costs, given by P(x) = 10x - 12 - 3x -
x^2

At break-even point P(x) =
0

=> 10x - 12 - 3x - x^2 =
0

=> x^2 - 7x + 12 =
0

=> x^2 - 4x - 3x + 12 =
0

=> x(x - 4) - 3(x - 4) =
0

=> (x - 3)(x - 4) =
0

x = 3 and x =
4

We find that if either 3 or 4 items are
manufactured the profit is zero. For all other values of x the profit is negative. There
is no profit made by producing and selling this item at the given terms (i.e. there is
no break-even point).