Answer:
x=8.75
Step-by-step explanation:
The price x that maximizes profit is the maximum value of the function, and the maximum value of the function is located at a point where the first derivative of the function is equal to zero. The first derivative is:
[tex]P(x) = - 2x^2+35x-99\\P'(x)=-2(2)x^{(2-1)}+35(1)-0\\P'(x)=-4x+35[/tex]
Using P'(x)=0:
[tex]0=-4x+35\\4x=35\\x=35/4\\x=8.75[/tex]
The minimum value of the function is also at a point where the first derivative of the function is equal to zero. To differentiate if x=8. is a minimum or a maximum obtain the second derivative and evaluate it at x=8.75 if the value P''(x)>0 x is minimum and if P''(x)<0 x is a maximum.
[tex]P'(x)=-4x+35\\P''(x)=-4(1)\\P''(x)=-4[/tex]
Evaluating at x=8.75:
[tex]P''(8.75)=-4[/tex]
Therefore, x=8.75 is the maximum value of the function and it is the price that maximizes profit.