Optimization Question about Profit, Revenue and Cost

Question

I have a question about this problem. I don't know what to do or where to start.

I have a function of R(x) and C(x), which are income earned/Revenue function and cost function respectively.

I also have a profit function, that is P(x) = R(x) - C(x)

The main question is that if the manufacturing is increasing to 52 smartphones per week, and their current output is 100 smartphones, what is the rate that their profit will be increasing?

What part do I need differentiation? How do I get the critical values? How do I get the rate?

Thank you very much.

Answer 1

P'(x) = R'(x)-C'(x)

rate of increase = P'(152)-P'(100) = R'(152) - C'(100) - R'(100) + C'(100)

take the derivative of the profit function to get the rate of increase = MR-MC = marginal revenue minus marginal cost = marginal profit

set that equal to zero to find crital values, such as the relative maximum or minimum profit

or

calculate P(152) - P(100) and divide by 52

rate of increase = [P(152)-P(100)]/52 = [R(152)-R(100) - C(152)+C(100)]/52

1st approach gives the rate of increase at a point

2nd approach gives the rate of increase over an interval

It would help if you had the specific functions to see what's going on better, algebraically and graphically. graph marginal revenue and marginal cost. Where they intersect is maximum profit (or minimum loss)