Answer:
A company determines that its marginal profit, in dollars, for
producing x units of a product is given by P'(x) = 6800x4,
where x > 1. Suppose it were possible for this company to make
infinitely many units of this product. What would the total profit be?
To find the total profit, we need to integrate the marginal profit function to obtain the profit function, and then evaluate the profit function at infinity.
The profit function is obtained by integrating the marginal profit function:
P(x) = ∫ P'(x) dx = ∫ 6800x^4 dx = 1360x^5 + C
where C is the constant of integration.
Since the company can produce infinitely many units, we can assume that they will continue to produce units until the marginal cost equals zero, which means that the marginal profit is also zero. This occurs at the maximum of the profit function.
To find the maximum of the profit function, we can take the derivative of the profit function with respect to x and set it equal to zero:
P'(x) = 6800x^4 = 0
x = 0
However, the condition given is x > 1, which means that the maximum occurs at the limit as x approaches infinity:
lim x→∞ P(x) = lim x→∞ (1360x^5 + C) = ∞
Therefore, the total profit is infinite if the company can produce infinitely many units of the product.