Tell the maximum number of zeros that the polynomial function may have. Then use Descartes' Rule of Signs to determine how many positive and how many negative zeros the polynomial function may have. Do not attempt to find the zeros. f(x) = 3x -x2 +x+8 What is the maximum number of zeros this polynomial function can have?
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 Zeros Calculation
**Objective:**
Determine the maximum number of zeros that the polynomial function may have. Then, utilize Descartes' Rule of Signs to ascertain the number of positive and negative zeros the polynomial function may possess. Do not attempt to find the zeros.
**Given Polynomial Function:**
\[ f(x) = 3x^3 - x^2 + x + 8 \]
**Step-by-Step Instructions:**
1. **Maximum Number of Zeros:**
The maximum number of zeros a polynomial function can have is equal to its degree. For the given polynomial \( f(x) = 3x^3 - x^2 + x + 8 \), the degree is 3. Therefore, the maximum number of zeros is 3.
2. **Descartes' Rule of Signs:**
- **Positive Zeros:**
Count the number of sign changes in \( f(x) = 3x^3 - x^2 + x + 8 \).
\[
\begin{align*}
3x^3 & \quad (- \text{sign change}) \\
- x^2 & \quad (+ \text{sign change}) \\
+ x & \quad (+ \text{no sign change}) \\
+ 8
\end{align*}
\]
There are 2 sign changes, implying there could be 2 or 0 positive zeros.
- **Negative Zeros:**
To find the negative zeros, evaluate \( f(-x) \).
\[
f(-x) = 3(-x)^3 - (-x)^2 + (-x) + 8 = -3x^3 - x^2 - x + 8
\]
Count the number of sign changes in \( f(-x) \).
\[
\begin{align*}
-3x^3 & \quad (- \text{no sign change}) \\
- x^2 & \quad (- \text{no sign change}) \\
- x & \quad (+ \text{sign change}) \\
+ 8
\end{align*}
\]
There is 1 sign change, implying there could be 1 or 0 negative zeros.
**Question:**
What is the maximum number of zeros this polynomial function can](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdafe61fc-c70a-4579-bf5c-a90ac61b494b%2F3e2454d6-1c95-4abc-a7c3-19c883ca9505%2Fvo990x_processed.png&w=3840&q=75)
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