A firm maximises profitt at 5 units, breakeven at 8 units, but loses 10 000 if nothing is sold, determine the quadratic function relating to units sold.

Starting with the general form of the quadratic equation:


y = ax^2 + bx + c


what you want to do is determine values for b and c in terms of a, then solve for a.

From the first condition you know that the derivative of y equals zero at x = 5


\frac {dy} {dx} = 2ax + b\\

Therefore

0 = 2a(5) + b\\
b = -10a

So now you can substitue for b in the basic quadratic equation.

The second piece of information you were given tells you that the value of y is 0 at x = 8, so:


0 = a \cdot 8^2 -10 \cdot 8 \cdot a +c


From this you can express c in terms of a.

Finally the last condition tells you that y(0) = -10000. From this you can now determine the value of a.

Hope this helps.