Give a possible formula of minimum degree for the polynomial h(x) displayed in the graph to the right. NOTE: Enter the exact answer. -1 -2 h(x) =
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.
![**Polynomial Functions and Graphs**
**Current Attempt in Progress**
**Problem Statement:**
*Give a possible formula of minimum degree for the polynomial \( h(x) \) displayed in the graph to the right. NOTE: Enter the exact answer.*
**Polynomial Input:**
\[ h(x) = \]
**Graph Explanation:**
The graph alongside the problem statement shows a polynomial function \( h(x) \) plotted on an \( xy \)-plane.
- **Axes:** The horizontal axis represents the \( x \)-values ranging from approximately \(-4\) to \( 4 \), and the vertical axis represents the \( y \)-values ranging from approximately \(-4\) to \( 4 \).
- **Curve Behavior:** The graph is a smooth, continuous curve crossing the \( x \)-axis at four points, indicating that these are the roots of the polynomial. These roots approximately occur at \( x = -3, -1, 1, 3 \).
- **Turning Points:** The graph has three turning points - it changes direction suggesting maximums and minimums between the roots.
**Interpretation Task:**
- Based on the graph, identify the roots of the polynomial and construct a possible polynomial formula with these roots.
- Consider the parities of the polynomial and deduce the correct factors that would fit the minimum degree.
**Possible Polynomial Construction:**
Given the roots observed at \( x = -3, -1, 1, 3 \), you can form the polynomial in the form:
\[ h(x) = k(x + 3)(x + 1)(x - 1)(x - 3) \]
where \( k \) is a leading coefficient which could be 1 or another constant depending on the specific vertical stretching or compressing of the polynomial graph.
Students are expected to observe the critical points and use their knowledge to enter the exact mathematical form of the polynomial accurately.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F660a15be-f1ca-4f1d-9222-efc14cd423b5%2F1b0754a5-f050-4650-b5eb-1b61868f19f9%2F91v4ac_processed.jpeg&w=3840&q=75)
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
