P = 3.44x – 0.00005 x^2– 5000, 0 ≤ x ≤ 67,300.The profit (in dollars) made by a fast-food restaurant selling x hamburgers isP = 3.44x – 0.00005 x^2– 5000, 0 ≤ x ≤ 67,300.Find the...

P = 3.44x – 0.00005 x^2– 5000, 0 ≤ x ≤ 67,300.

The profit (in dollars) made by a fast-food restaurant selling x hamburgers is
P = 3.44x – 0.00005 x^2– 5000, 0 ≤ x ≤ 67,300.
Find the open intervals on which P is increasing or decreasing.

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You should remember that the first derivative of function helps you to find the intervals where the function increases or decreases.

You should find the first derivative such that:

`P'(x) = (3.44x - 0.00005 x^2 -5000)'`

`P'(x) = 3.44 - 0.0001x`

You need to determine the zero of `P'(x)`  such that:

`P'(x) = 0 =gt 3.44 - 0.0001x = 0 =gt x = 3.44/0.0001`

`x = 34400`

Hence, the values of P'(x) are positive if `x in (0,34400) ` and negative if `x in (34400 , 67300), ` hence, the function increases over (0,34400) and it decreases over (34400 , 67300).

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