The profit function P(x) is a parabola.
To find the maximum profit, you need the vertex of the parabola.
The vertex can be found using the formula [tex]x = \frac{-b}{2a} [/tex]
where a = -.001 , b = 3, c = -1800
[tex]x = \frac{-3}{2(-.001)} = 1500[/tex]
Part a) Profit is maximized when x = 1500.
To find Profit, plug 'x' into profit function.
[tex]P_{max} = -.001(1500)^2 + 3(1500) - 1800 = 450[/tex]
Part b) Daily max Profit is $450
Next to find the break even point, when profit = 0.
Set profit function equal to 0 and solve for 'x'.
This is a quadratic equation, so use the quadratic formula.
[tex]x = \frac{-b \pm \sqrt{b^2 -4ac}}{2a} \\ \\ x = \frac{-3 \pm \sqrt{9 - 4(-.001)(-1800)}}{2(-.001)} \\ \\ x = 829.18 , 2170.82[/tex]
Finally, the avg rate of change is simply the slope of the line between 2 points on the parabola. Use slope formula.
[tex]= \frac{P(2100) - P(1200)}{2100 - 1200} = \frac{90 - 360}{2100-1200} = \frac{-270}{900} = -\frac{3}{10} [/tex]
Part d) Avg rate of change = -3/10
