A lot of companies use the marginal cost function for their business models to determine the production level to minimize total costs. As we all know, if C(x) represents the total cost of producing x units of a product, then the average cost per unit at this production level is C(x)/x . Let assume C(x) is differentiable on all x>0 . This implies that the average cost function is also differentiable for all x>0 since C(x)/x can be written as C(x)⋅1/x , and both C(x) and 1/x are differentiable for all x>0. Suppose you go to an interview and the chairman of the interview committee gives you the following questions: In our company , we use marginal cost to determine the production level to minimize the costs. However, there is another useful model in business, the average cost function, that we also track. Is it true or not that the marginal cost C′(x) is equal to the average cost at the critical points of the average cost function? Please elaborate and explain . This year, the costs at the x production level is expected to be C(x)=16000+200x+4x^3/2 , in dollars. What production level would you advise us to hold in order to minimize the average cost per item?
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.
A lot of companies use the marginal cost function for their business models to determine the production level to minimize total costs. As we all know, if C(x) represents the total cost of producing x units of a product, then the average cost per unit at this production level is C(x)/x . Let assume C(x) is differentiable on all x>0 . This implies that the average cost function is also differentiable for all x>0 since C(x)/x can be written as C(x)⋅1/x , and both C(x) and 1/x are differentiable for all x>0.
Suppose you go to an interview and the chairman of the interview committee gives you the following questions:
In our company , we use marginal cost to determine the production level to minimize the costs. However, there is another useful model in business, the average cost function, that we also track.
- Is it true or not that the marginal cost C′(x) is equal to the average cost at the critical points of the average cost function? Please elaborate and explain .
- This year, the costs at the x production level is expected to be C(x)=16000+200x+4x^3/2 , in dollars. What production level would you advise us to hold in order to minimize the average cost per item?
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