The quantity (in lbs) of a coffee that is sold by a company at a price of p dollars / lb is Q = ƒ(x).?

Question

What is the meaning of the derivative ƒ'(8)? What are it's units?

Is ƒ'(8) positive or negative? Explain.

The question is indeed supposed to read ƒ(p), my apologies.

Answer 1

The derivative talks about the rate at which "pounds sold" changes as the price goes up. f'(8) says, if the price is currently $8/lb and we raise the price by one (tiny) unit, how much do we expect sales to change?

Common sense suggests that the more you charge for something, the fewer people will buy it, so f'(8) is probably negative. On the other hand, I heard of a company being told by consultants that if they raise their price, customers will have more respect for their brand name--so more people will buy their product. That seems to fly in the face of the principal of supply and demand, but it's so crazy it just might work.

If f'(8) is positive (corresponding to this strange scenario), then the best move for economic advantage is to raise the price. If f'(8) is negative, the decision is a bit more involved.

By the way, f'(x) represents a change in pounds sold over a change in dollars per pound, or (lbs.)/(($/lb)), which works out to (lb)^2/$ (square pounds per dollar). What's a square pound? Uhm...

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Edit: after reading the above answer, it occurred to me that you said p dollars per pound but used x in your function. If those are supposed to be the the same variable (both p or both x), then it's as I said. If x represents some unit of labor, then the answer will be more like the previous answer.