The function f is decreasing over which intervals? Choose all that apply. (-00 , -4) O (-4, 0) O (0, 4) O (4, 7) O (o, 7) O (7, 00 ) (b) The function f has local maxima at which x-values? If there is more than one value, separate them with commas.
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.
Problem PageQuestion Below is the graph of a polynomial function with real coefficients. Use the graph to answer the following questions about . All local extrema of are shown in the graph.
![The image contains a graph of a function and a set of related questions. Below is a detailed description and explanation suitable for an educational website:
---
### Graph Analysis of Function \( f \)
The graph provided illustrates the behavior of the function \( f \). The horizontal axis represents the \( x \)-values, while the vertical axis represents the \( y \)-values of the function.
- **Graph Description:** The function \( f \) shows a curve that starts at a high value, dips down, rises to form a peak, lowers again to a valley, and then increases as \( x \) continues.
#### Questions and Answers
**(a) The function \( f \) is decreasing over which intervals? Choose all that apply.**
- \[
\boxed{ }
\]
Options:
- \( (-\infty, -4) \)
- \( (-4, 0) \)
- \( (0, 4) \)
- \( (4, 7) \)
- \( (7, \infty) \)
**(b) The function \( f \) has local maxima at which \( x \)-values? If there is more than one value, separate them with commas.**
- \[
\boxed{ }
\]
**(c) What is the sign of the leading coefficient of \( f \)?**
- \[
\boxed{\text{Select One} }
\]
**(d) Which of the following is a possibility for the degree of \( f \)? Choose all that apply.**
- \[
\boxed{ }
\]
Options:
- 4
- 5
- 6
- 7
- 8
- 9
---
This information, along with the questions, helps in understanding the properties and behavior of the function \( f \) as depicted in the graph. The learners are to identify the intervals where the function is decreasing, determine the local maxima points, establish the sign of the leading coefficient, and suggest possible degrees of the function \( f \) based on the given options.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F64f54e3f-bbb2-46db-a1a9-bb995302d054%2F14bc6e45-5dc7-43b3-9e9f-3590dc4c62cf%2F0rgwl0f.png&w=3840&q=75)
Trending now
This is a popular solution!
Step by step
Solved in 6 steps with 9 images
