Sketch the graph of the following function. List the coordinates of where extrema or points of inflection occur. State where the function is increasing or decreasing as well as where it is concave up or concave down. f(x) = 2x° - 24x – 8 On what interval(s) is f increasing or decreasing? O A. fis increasing on (- 0,0). fis decreasing on (0,00). O B. fis increasing on (-00,00). fis never decreasing. OC. fis increasing on (-2,2). f is decreasing on (- 0, – 2) and (2,00) O D. fis increasing on (0,00). f is decreasing on (- 00,0). O E. fis never increasing. f is decreasing on (- 00,00). OF. fis increasing on (- 00, - 2) and (2,00). f is decreasing on (-2,2). On what interval(s) is f concave up or concave down? O A. fis concave up on (-o0, - 2) and (2,00). f is concave down on (-2,2). O B. fis concave up on (-oc0,0) and (0,00). f is never concave down. O c. fis concave up on (0,00). fis concave down on (-o,0). O D. fis concave up (-2,2). f is concave down on (- 00, -2) and (2,00). O E. fis never concave up. f is concave down on (- 00,0) and (0,00). O F. fis concave up on (- a0,0). fis concave down on (0,00).
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.
![### Function Analysis and Graph Sketching
**Given Function:**
\[ f(x) = 2x^3 - 24x - 8 \]
**Tasks:**
1. Determine the intervals on which the function is increasing or decreasing.
2. Identify the intervals of concavity (concave up or concave down).
#### 1. Intervals of Increase and Decrease
**On what interval(s) is \( f \) increasing or decreasing?**
- **Option A:**
- \( f \) is increasing on \( (-\infty, 0) \)
- \( f \) is decreasing on \( (0, \infty) \)
- **Option B:**
- \( f \) is increasing on \( (0, \infty) \)
- \( f \) is never decreasing
- **Option C:**
- \( f \) is increasing on \( (-2, 2) \)
- \( f \) is decreasing on \( (-\infty, -2) \) and \( (2, \infty) \)
- **Option D:**
- \( f \) is increasing on \( (0, \infty) \)
- \( f \) is decreasing on \( (-\infty, 0) \)
- **Option E:**
- \( f \) is never increasing
- \( f \) is decreasing on \( (-\infty, \infty) \)
- **Option F:**
- \( f \) is increasing on \( (-\infty, -2) \) and \( (2, \infty) \)
- \( f \) is decreasing on \( (-2, 2) \)
#### 2. Intervals of Concavity
**On what interval(s) is \( f \) concave up or concave down?**
- **Option A:**
- \( f \) is concave up on \( (-\infty, -2) \) and \( (2, \infty) \)
- \( f \) is concave down on \( (-2, 2) \)
- **Option B:**
- \( f \) is concave up on \( (-\infty, 0) \)
- \( f \) is never concave down
- **Option C](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc48c62e0-319f-4142-a578-7bc904cc021f%2F4a2ba57d-cdb2-4e82-ae9e-4a312c3e964f%2Ffvcq3i_processed.png&w=3840&q=75)
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