The function that represents this company’s profit f(x) depends on the number of items sold is [tex]\rm f(x)=-2(x-105)^2+18,050[/tex] and it can be determined by using properties of function.
Given that,
According to one company’s profit model, the company has a profit of 0 when 10 units are sold and a maximum profit of $18,050 when 105 units are sold.
We have to determine,
What is the function that represents this company’s profit f(x) depending on the number of items sold, x?
According to the question,
Let, x be the amount sold,
According to one company’s profit model, the company has a profit of 0 when 10 units are sold.
Then,
The function represents the company has a profit of 0 when 10 units are sold is,
[tex]\rm f(x) = profit\\\\f(10) =0[/tex]
The maximum profit of $18,050 when 105 units are sold.
Therefore,
The function represents the maximum profit of $18,050 when 105 units are sold is,
[tex]\rm f(105) = 18050[/tex]
Differentiate the function
[tex]\rm f'(105)=0[/tex]
[tex]\rm \ And \ f''(105) = 0\\[/tex]
The function which satisfies all the conditions is,
[tex]\rm f(x)=-2(x-105)^2+18,050[/tex]
Hence, The function that represents this company’s profit f(x) depends on the number of items sold is [tex]\rm f(x)=-2(x-105)^2+18,050[/tex].
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