4) thousand units sold/ thousand units sold per week/ thousand units sold per week per week. 5) a. where the function equals zero/ the critical points, where the derivative is zero/ the inflection points, where the second derivative is zero b. function value/ critical point/ inflection point c. Relative maximum/ relative minimum d. Concave up/ concave down e. Relative maximum/ relative minimum f. Concave up/concave down
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.
![**3) Which of the following is the most appropriate choice for the second derivative, \( P''(t) \)**
- **A) \( P''(t) = 0 \)**
- **B) \( P''(t) = -0.0786t + 1.322 \)**
- **C) \( P''(t) = -0.0786t + 1.322 - 2.021 \)**
**Option [Select] is the most appropriate choice for \( P''(t) \).**
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**4) The units for the second derivative, \( P''(t) \), would be [Select] .**
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**5) To find the optimal value we would first find [Select] . If using the second derivative test, if the value of the second derivative at the [Select] is positive then there is a [Select] because the graph is [Select] , if it was negative then there is a [Select] since the graph is [Select] .**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F73c45e44-e5f7-43e8-a1b3-3500cc988f4c%2F16e8199d-57bf-4854-8a2c-c80848d03586%2Foscykj6_processed.jpeg&w=3840&q=75)
![### Derivative Problem Set for Sales Model
**Problem Statement:**
For each question, choose the most appropriate answer.
The model for the sales of a product in thousands of units is given by
\[ S(t) = -0.0131t^3 + 0.661t^2 - 2.021t + 7.876 \]
where \( t \) is in weeks after the release of the product.
1) Which of the following is the most appropriate choice for the derivative, \( P'(t) \)?
**Options:**
A) \( P'(t) = -0.0393t^2 + 1.322t - 2.021 \)
B) \( P'(t) = -0.0131t^2 + 0.661x - 2.021 \)
C) \( P'(t) = 0 \)
**Instruction:**
Select the option which is the most appropriate choice for \( P'(t) \).
**Note:** The correct choice reflects the derivative of the sales model \( S(t) \) with respect to time \( t \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F73c45e44-e5f7-43e8-a1b3-3500cc988f4c%2F16e8199d-57bf-4854-8a2c-c80848d03586%2Fppq2cpm_processed.jpeg&w=3840&q=75)
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