6. DETAILS A polynomial function is given. P(x) = x(x - 4) (a) Describe the end behavior of the polynomial function. End behavior: as x → 00 as x → -c∞ (b) Match the polynomial function with one of the following graphs. y y 10 10 5 5- -3 2 -1 1 -3 -5 -5- -10 -10 y
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 6: Polynomial Function Analysis**
A polynomial function is given as
\[ P(x) = x(x^2 - 4) \]
### (a) Describe the end behavior of the polynomial function.
To determine the end behavior, analyze how the polynomial behaves as \( x \) approaches positive infinity (\( \infty \)) and negative infinity (\( -\infty \)):
\[ \text{End behavior:} \]
\[ \text{As } x \rightarrow \infty, \quad y \rightarrow \boxed{\;\, } \]
\[ \text{As } x \rightarrow -\infty, \quad y \rightarrow \boxed{\;\, } \]
### (b) Match the polynomial function with one of the following graphs.
Below are two graphs provided to visually assess which one corresponds to the polynomial \( P(x) = x(x^2 - 4) \).
**Graph Analysis**:
- **Graph on the Left:**
- The graph intersects the x-axis at three points: approximately \(-2\), \(0\), and \(2\).
- The graph descends towards negative infinity as \( x \) approaches positive and negative infinity.
- The function has local maxima and minima, displaying a cubic behavior.
- **Graph on the Right:**
- The graph intersects the x-axis at two points: approximately \(-2\) and \(2\).
- The ends of the graph rise towards positive infinity as \( x \) approaches positive and negative infinity.
- The graph has a parabolic shape and does not match the expected cubic behavior.
Thus, the correct match for the polynomial function \( P(x) = x(x^2 - 4) \) is the graph on the **left**.
Select the graph corresponding to the function \( P(x) = x(x^2 - 4) \):
\[ \boxed{\text{Left Graph}} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f4719ec-89c6-49f2-a8fd-c905fe776b01%2Fe0308bc7-ab71-4499-926a-d444f535a958%2F3g56c2_processed.jpeg&w=3840&q=75)
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