The production function f (x) gives the number of units of an item that a manufacturing com pany can produce from units of raw material. The company buys the raw material at price w dollars per unit and sells all it produces at a price of p dollars per unit. The quantity of raw material that maximizes profit is denoted by x*.

(a) Do you expect the derivative f' (x) to be positive or negative? Justify your answer

(b) Explain why the formula π (x) = pf (x) - wx gives the profit π (x) that the company earns as a function of the quantity x of raw materials that it uses.

(c) Evaluate f' (x*).

Step-by-step explanation:

Recall that the notion of the derivative of a function is the rate of change of it. So it kind of tells us how much the value of functioin changes as the independt variable increases or decreases. If it is positive, this means that the function will increase as the indepent variable increases, and if it is negative, that means that the function will decrease as the indepent variable increases.

a) Since f (x) is the number of units you can make out of x units of raw material, it is natural to think that the more material you have, the more units you can make, so we expect f' (x) to be positive.

b) The company buys each unit of raw material at the price w. So the product wx represents the total cost of the raw material used to produce f (x) units. Since each produced unit is sell at the price of p, then the product pf (x) represents the total income for selling all f (x) units. Recall that the profit is the difference between the total income and the total cost of production. Hence, the profit in this case is represented by the formula pf (x) - wx.

c) Recall that a function h (x) that is differentiable attains it's maximum when it's derivative is 0 and it's second derivative is negative.

In this case, we know that the derivative of the profit function, evaluated at x * must be 0, since it is a maximum. So, using the rules of derivation, we know that the derivative of the profit function is pf' (x) - w. Hence,

pf' (x*) - w = 0. From where we know that f' (x*) = w/p.