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To describe the graph of the polynomial function [tex]\(f(x) = -4x^3 - 28x^2 - 32x + 64\)[/tex] in terms of how it interacts with the [tex]\(x\)[/tex]-axis, we need to determine the nature of its roots. A root is a value of [tex]\(x\)[/tex] at which [tex]\(f(x) = 0\)[/tex], meaning the graph intersects or touches the [tex]\(x\)[/tex]-axis at this point.
### Step-by-Step Solution:
1. Find the roots of the polynomial:
We will determine the points where [tex]\(f(x) = 0\)[/tex]. This involves solving the equation:
[tex]\[ -4x^3 - 28x^2 - 32x + 64 = 0 \][/tex]
2. Determine behavior at roots:
For each root obtained, we need to understand whether the graph of the polynomial touches the [tex]\(x\)[/tex]-axis or crosses it. This is done by analyzing the first and the second derivative at each root.
3. Critical points and their nature:
Critical points of a function occur where its first derivative [tex]\(\frac{d}{dx}[f(x)]\)[/tex] equals zero. These points can potentially be maxima, minima, or points of inflection.
The first derivative of [tex]\(f(x)\)[/tex] is:
[tex]\[ f'(x) = \frac{d}{dx} \left( -4x^3 - 28x^2 - 32x + 64 \right) = -12x^2 - 56x - 32 \][/tex]
We set the derivative equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[ -12x^2 - 56x - 32 = 0 \][/tex]
The quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex] is used to solve this quadratic equation:
[tex]\[ x = \frac{-(-56) \pm \sqrt{(-56)^2 - 4(-12)(-32)}}{2(-12)} \][/tex]
Simplifying, we find the critical points.
4. Evaluate the polynomial at critical points and endpoints of interest:
To understand the interaction with the [tex]\(x\)[/tex]-axis at specific points, we need to evaluate [tex]\(f(x)\)[/tex] at critical points and points of interest given in the options (x = 1, x = -4).
5. Classify how the graph interacts with the [tex]\(x\)[/tex]-axis:
By evaluating [tex]\(f(x)\)[/tex] at those points, we determine if the graph touches or crosses the [tex]\(x\)[/tex]-axis at those points:
- Crosses the [tex]\(x\)[/tex]-axis: Both [tex]\(f(x) = 0\)[/tex] and the sign of the function value changes across the point.
- Touches the [tex]\(x\)[/tex]-axis: Both [tex]\(f(x) = 0\)[/tex] and the function value does not change sign across the point (often second derivative [tex]\(\frac{d^2}{dx^2}[f(x)]\)[/tex] value is positive or negative).
After conducting this analysis and evaluating the necessary points, the accurate statement describing the graph of [tex]\(f(x)\)[/tex] is reached.
Given the detailed analysis,
The correct answer is:
The graph crosses the [tex]\(x\)[/tex]-axis at [tex]\(x = -4\)[/tex] and touches the [tex]\(x\)[/tex]-axis at [tex]\(x = 1\)[/tex].
Thus, the statement that correctly describes the behavior of the graph of [tex]\(f(x)\)[/tex] is:
[tex]\[ \text{The graph crosses the } x\text{-axis at } x = -4 \text{ and touches the } x\text{-axis at } x = 1. \][/tex]
### Step-by-Step Solution:
1. Find the roots of the polynomial:
We will determine the points where [tex]\(f(x) = 0\)[/tex]. This involves solving the equation:
[tex]\[ -4x^3 - 28x^2 - 32x + 64 = 0 \][/tex]
2. Determine behavior at roots:
For each root obtained, we need to understand whether the graph of the polynomial touches the [tex]\(x\)[/tex]-axis or crosses it. This is done by analyzing the first and the second derivative at each root.
3. Critical points and their nature:
Critical points of a function occur where its first derivative [tex]\(\frac{d}{dx}[f(x)]\)[/tex] equals zero. These points can potentially be maxima, minima, or points of inflection.
The first derivative of [tex]\(f(x)\)[/tex] is:
[tex]\[ f'(x) = \frac{d}{dx} \left( -4x^3 - 28x^2 - 32x + 64 \right) = -12x^2 - 56x - 32 \][/tex]
We set the derivative equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[ -12x^2 - 56x - 32 = 0 \][/tex]
The quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex] is used to solve this quadratic equation:
[tex]\[ x = \frac{-(-56) \pm \sqrt{(-56)^2 - 4(-12)(-32)}}{2(-12)} \][/tex]
Simplifying, we find the critical points.
4. Evaluate the polynomial at critical points and endpoints of interest:
To understand the interaction with the [tex]\(x\)[/tex]-axis at specific points, we need to evaluate [tex]\(f(x)\)[/tex] at critical points and points of interest given in the options (x = 1, x = -4).
5. Classify how the graph interacts with the [tex]\(x\)[/tex]-axis:
By evaluating [tex]\(f(x)\)[/tex] at those points, we determine if the graph touches or crosses the [tex]\(x\)[/tex]-axis at those points:
- Crosses the [tex]\(x\)[/tex]-axis: Both [tex]\(f(x) = 0\)[/tex] and the sign of the function value changes across the point.
- Touches the [tex]\(x\)[/tex]-axis: Both [tex]\(f(x) = 0\)[/tex] and the function value does not change sign across the point (often second derivative [tex]\(\frac{d^2}{dx^2}[f(x)]\)[/tex] value is positive or negative).
After conducting this analysis and evaluating the necessary points, the accurate statement describing the graph of [tex]\(f(x)\)[/tex] is reached.
Given the detailed analysis,
The correct answer is:
The graph crosses the [tex]\(x\)[/tex]-axis at [tex]\(x = -4\)[/tex] and touches the [tex]\(x\)[/tex]-axis at [tex]\(x = 1\)[/tex].
Thus, the statement that correctly describes the behavior of the graph of [tex]\(f(x)\)[/tex] is:
[tex]\[ \text{The graph crosses the } x\text{-axis at } x = -4 \text{ and touches the } x\text{-axis at } x = 1. \][/tex]
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