Tell the maximum number of zeros that the polynomial function may have. Then use real zeros the polynomial function may have. Do not attempt to find the zeros: 8. f(x) = - 5x° +2x° -6 What is the maximum number of zeros that this polynomial function can have? How many positive real zeros can the function have? (Use a comma to separate answers as needed.) How many negative real zeros can the function have? (Use a comma to separate answers as needed.)
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 Function Analysis**
In this exercise, you are tasked with determining information about the zeros of a polynomial function. The given polynomial is:
\[ f(x) = -5x^8 + 2x^6 - 6 \]
### Questions to Consider:
1. **What is the maximum number of zeros that this polynomial function can have?**
- The degree of the polynomial informs the maximum possible number of zeros. For this polynomial, the degree is 8, meaning it can have up to 8 zeros.
2. **How many positive real zeros can the function have?**
- Use Descartes' Rule of Signs to determine the potential number of positive real zeros. List your answers with commas separating any possibilities.
3. **How many negative real zeros can the function have?**
- Again, use Descartes' Rule of Signs to determine the potential number of negative real zeros. Provide your answers separated by commas.
**Note:** Please enter your answers in each of the provided answer boxes. Descartes' Rule of Signs helps predict the number of positive and negative real zeros based on sign changes. Do not attempt to find the exact zeros; focus on determining the possibilities based on the function’s characteristics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F323a188d-72c2-4f77-bbb3-3034e58495c9%2F65abb5f8-5af4-4781-a3d5-dec5543df6cc%2Fzhm9dbb_processed.jpeg&w=3840&q=75)
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