This can be determined by using first derivative test.Therefore, her revenue cannot be at least $25,000 this year.
Given :
Revenue is given by the function as R = [tex]\bold{-0.5x^{2} -200x}[/tex] ...(1)
By using of first derivative test, Margaret revenue will be at the maximum if
[tex]\bold{\dfrac{dR}{dx} =0}[/tex]
By first derivative test,
[tex]\dfrac{dR}{dx} =-2(0.5)x -200\\\dfrac{dR}{dx}= -x -200\\[/tex] ...(2)
As we know that ,
[tex]\dfrac{dR}{dx} =0[/tex]
Now, calculate further from equation (2),
[tex]\begin{aligned}-&x-200=0\\\bold{&x=-200}\end{aligned}[/tex]
Now, put the value of [tex]\bold{x=-200}[/tex] in equation (1) and solve it further,
[tex]R&=0.5x^{2} -200x\\R&=0.5(-200)^{2}-200(-200)\\R &= -0.5(40000)+40000R = -20000+40000R = 20000[/tex]
Therefore,it means that the revenue for the year is $20,000. From this we have to show that her revenue cannot be at least $25000 for that year,but she could make less than $25000.
For further details, please refer this link:
https://brainly.com/question/16882582
