Answer:
Selling 40 items will produce a maximum profit.
Step-by-step explanation:
We need to use First and Second Derivative Tests on profit function to determine how many items will lead to maximum profit. Let [tex]p(x) = -x^{3}+30\cdot x^{2}+2400\cdot x -26000[/tex], where [tex]p(x)[/tex] is the profit for a product, measured in US dollars, and [tex]x[/tex] is the amount of items, dimensionless.
First we derive the profit function and equalize it to zero:
[tex]p'(x) = -3\cdot x^{2}+60\cdot x+2400[/tex]
[tex]-3\cdot x^{2}+60\cdot x +2400 = 0[/tex] (Eq. 1)
Roots are found by Quadratic Formula:
[tex]x_{1} = 40[/tex] and [tex]x_{2} = -20[/tex]
Only the first root may offer a realistic solution. The second derivative of the profit function is found and evaluated at first root. That is:
[tex]p''(x) = -6\cdot x +60[/tex] (Eq. 2)
[tex]p''(40) = -6\cdot (40)+60[/tex]
[tex]p''(40) = -180[/tex] (Absolute maximum)
Therefore, selling 40 items will produce a maximum profit.
