For a certain company, the cost function for producing x items is C(x)=40x+100 and the revenue function for selling x items is R(x)=−0.5(x−100)2+5,000. The maximum capacity of the company is 150 items. The profit function P(x) is the revenue function R(x) (how much it takes in) minus the cost function C(x) (how much it spends). In economic models, one typically assumes that a company wants to maximize its profit, or at least make a profit! Answers to some of the questions are given below so that you can check your work. Assuming that the company sells all that it produces, what is the profit function? P(x)= . Hint: Profit = Revenue - Cost as we examined in Discussion 3. What is the domain of P(x)? Hint: Does calculating P(x) make sense when x=−10 or x=1,000? The company can choose to produce either 60 or 70 items. What is their profit for each case, and which level of production should they choose? Profit when producing 60 items = Profit when producing 70 items = Can you explain, from our model, why the company makes less profit when producing 10 more units?
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
For a certain company, the cost function for producing x items is C(x)=40x+100 and the revenue function for selling x items is R(x)=−0.5(x−100)2+5,000. The maximum capacity of the company is 150 items.
The profit function P(x) is the revenue function R(x) (how much it takes in) minus the cost function C(x) (how much it spends). In economic models, one typically assumes that a company wants to maximize its profit, or at least make a profit!
Answers to some of the questions are given below so that you can check your work.
- Assuming that the company sells all that it produces, what is the profit function?
P(x)= .
Hint: Profit = Revenue - Cost as we examined in Discussion 3.
- What is the domain of P(x)?
Hint: Does calculating P(x) make sense when x=−10 or x=1,000?
- The company can choose to produce either 60 or 70 items. What is their profit for each case, and which level of production should they choose?
Profit when producing 60 items =
Profit when producing 70 items =
- Can you explain, from our model, why the company makes less profit when producing 10 more units?
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