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.
![### Understanding Profit Changes and Sales Units
#### Economic Concepts and Calculations
In this section, we will explore some fundamental concepts related to profit changes and unit sales. Pay close attention to the following tasks and how to approach them:
#### Part d: Rate of Change in Profit
To determine the rate at which profit changes on average as the number of units sold increases, consider this specific scenario:
- The number of units sold rises from 90 to 180 units.
The question asks for the average rate of change in profit per unit increase. Use the formula for the average rate of change:
\[ \text{Average Rate of Change} = \frac{\Delta \text{Profit}}{{\Delta \text{Quantity}}} \]
where:
- \(\Delta \text{Profit}\) is the change in profit.
- \(\Delta \text{Quantity}\) is the change in the number of units sold (which in this case is from 90 to 180 units).
You need to enter the calculated value in the following box:
\[ \boxed{\phantom{0}} \ \text{dollars per unit} \]
#### Part e: Number of Units for Profit Maximization
This part requires identifying the number of units sold at the point where the profit stops increasing and begins to decrease. This typically involves finding:
- The maximum profit point, which may require calculus (finding the derivative of the profit function and setting it to zero) or analyzing given profit data.
If necessary, round your final answer to the nearest whole number. Enter this value in the designated box:
\[ \boxed{\phantom{0}} \ \text{units} \]
By following these steps and practicing with real data or hypothetical profit functions, students can gain a deeper understanding of economic and math principles related to business and sales optimization.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbf7777fa-aa3c-4ddb-8a8c-67588cdbbf49%2F89f7e111-3f41-4760-a522-695bc3c718c0%2Feve0hdf_processed.jpeg&w=3840&q=75)
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