1. Suppose C(x) is a function representing the cost (in dollars) of producing x units of energy, and R(x) is a function representing the revenue (in dollars) of selling x units of energy. Suppose further that both functions are continuous for all x > = 0.

Question

A. Is there necessarily some x value, let's call it C, that will maximize profit over all x > = 0? Explain your answer.

B. Suppose we know we can't produce more than 1,000 units of energy. Is there necessarily some x-value, call it C, that will maximize profit over the interval [0,1000]? Explain your answer.

C. Again, assume that we can't produce more than 1,000 units of energy. Is there necessarily some x value, let's call it C, for which the profit is exactly 0? Explain your answer

Answer 1

A. It's confusing to have both C(x) and C as a value.
Clearly it is possible to have P(x) = R(x) - C(x) that has no maximum.
For example, if both are linear functions.

B. Of course. P(x) is continuous, so it must have a maximum value on a closed interval, even if it is constant. That would be the maximum (and also minimum) value.

C. of course not. See A above

Answer 2

Thank you so much this really helped me!!!!