Use Descartes' rule of signs to determine the possible number of positive and negative real zeros, counting multiplicities, of the function, f(x) = -3x³ + 2x4 + x³ - 7x2 - 3x + 3
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.
![### Polynomials and Zeros: Using Descartes’ Rule of Signs and Calculator Zero Function
#### Determining Possible Real Zeros Using Descartes’ Rule of Signs
To determine the possible number of positive and negative real zeros of a given polynomial function, we employ Descartes’ Rule of Signs. Consider the function:
\[ f(x) = -3x^5 + 2x^4 + x^3 - 7x^2 - 3x + 3 \]
Steps to determine the possible positive zeros:
1. Identify the sign changes in the coefficients of \( f(x) \).
- The signs of the coefficients in \( f(x) \) are: -, +, +, -, -, +.
- This results in four sign changes (from - to +, + to -, - to +).
According to Descartes’ Rule of Signs, the number of positive real zeros is either equal to the number of sign changes or less than that by an even number. This function has up to 4 positive zeros.
Steps to determine the possible negative zeros:
1. Substitute \( x \) with \( -x \) to determine \( f(-x) \):
\[ f(-x) = -3(-x)^5 + 2(-x)^4 + (-x)^3 - 7(-x)^2 - 3(-x) + 3 \]
\[ = 3x^5 + 2x^4 - x^3 - 7x^2 + 3x + 3 \]
2. Identify the sign changes in the coefficients of \( f(-x) \).
- The signs of the coefficients in \( f(-x) \) are: +, +, -, -, +, +.
- This results in two sign changes.
Thus, the number of negative real zeros is either 2 or 0.
#### Finding Real Zeros Using a Calculator
Using a graphing calculator's Zero function, we can find the exact values of the real zeros of another polynomial function:
\[ f(x) = -15x^4 - 5x^2 + 2x + 1 \]
Steps to find the real zeros:
1. Input the function into the calculator.
2. Adjust the calculator’s window settings to ensure the graph of the function is visible.
3. Use the Zero function to identify where the graph intersects the x-axis](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F07b00a25-4bbc-4603-8563-821c16d74641%2F7de1ecb7-5987-49cf-81c5-ee489fec0cf8%2F4lxn2xkb_processed.jpeg&w=3840&q=75)
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