We have been given that the profit that the vendor makes per day by selling x pretzels is given by the function [tex]P(x)=-4x^2+2400x-400[/tex]. We are asked to find the number of pretzels that must be sold to maximize profit.
First of all, we will find the derivative of our given function.
[tex]P'(x)=-4\cdot 2x^{2-1}+2400[/tex]
[tex]P'(x)=-8x+2400[/tex]
Now, we will find the critical point by equating derivative with 0 as:
[tex]-8x+2400=0[/tex]
[tex]-8x+2400-2400=0-2400[/tex]
[tex]-8x=-2400[/tex]
[tex]\frac{-8x}{-8}=\frac{-2400}{-8}[/tex]
[tex]x=300[/tex]
Therefore, the company must sell 300 pretzels to maximize profit and option B is the correct choice.
