The latest demand equation for your Yoda vs. Alien T-shirts is given by q = −50x + 900 where q is the number of shirts you can sell in one week if you charge $x per shirt. The Student Council charges you $200 per week for use of their facilities, and the T-shirts cost you $7 each. Find the weekly cost as a function of the unit price x. C(x) = Hence, find the weekly profit as a function of x. P(x) = Determine the unit price you should charge to obtain the largest possible weekly profit.x = $ per T-shirtWhat is the largest possible weekly profit?$
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.
The latest demand equation for your Yoda vs. Alien T-shirts is given by
where q is the number of shirts you can sell in one week if you charge $x per shirt. The Student Council charges you $200 per week for use of their facilities, and the T-shirts cost you $7 each. Find the weekly cost as a function of the unit price x.
| C(x) = |
|
Hence, find the weekly profit as a function of x.
| P(x) = |
|
Determine the unit price you should charge to obtain the largest possible weekly profit.
x = $ per T-shirt
What is the largest possible weekly profit?
$
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