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To determine which statement correctly describes the graph of the function [tex]\( f(x) = -x^4 + 3x^3 + 10x^2 \)[/tex], we need to analyze its behavior at the points where it intersects or touches the [tex]\( x \)[/tex]-axis. Specifically, we need to find the [tex]\( x \)[/tex]-intercepts and determine whether each is a point where the graph crosses the [tex]\( x \)[/tex]-axis or only touches it.
Here is the step-by-step process:
1. Find the [tex]\( x \)[/tex]-intercepts: Solve [tex]\( f(x) = 0 \)[/tex].
Given the function [tex]\( f(x) = -x^4 + 3x^3 + 10x^2 \)[/tex], we solve:
[tex]\[ -x^4 + 3x^3 + 10x^2 = 0 \][/tex]
Factor the equation:
[tex]\[ x^2(-x^2 + 3x + 10) = 0 \][/tex]
This gives us two factors:
[tex]\[ x^2 = 0 \quad \text{and} \quad -x^2 + 3x + 10 = 0 \][/tex]
Solving [tex]\( x^2 = 0 \)[/tex]:
[tex]\[ x = 0 \][/tex]
To solve [tex]\( -x^2 + 3x + 10 = 0 \)[/tex], we can use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = -1 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = 10 \)[/tex]:
[tex]\[ x = \frac{-3 \pm \sqrt{3^2 - 4(-1)(10)}}{2(-1)} = \frac{-3 \pm \sqrt{9 + 40}}{-2} = \frac{-3 \pm \sqrt{49}}{-2} = \frac{-3 \pm 7}{-2} \][/tex]
This gives:
[tex]\[ x = \frac{-3 + 7}{-2} = \frac{4}{-2} = -2 \][/tex]
and
[tex]\[ x = \frac{-3 - 7}{-2} = \frac{-10}{-2} = 5 \][/tex]
Thus, the roots are [tex]\( x = 0, x = 5, \)[/tex] and [tex]\( x = -2 \)[/tex].
2. Determine the behavior at each intercept: Check whether the graph touches or crosses the [tex]\( x \)[/tex]-axis at each intercept by analyzing the first and second derivatives at those points.
- First and Second Derivatives:
[tex]\[ f'(x) = \frac{d}{dx}(-x^4 + 3x^3 + 10x^2) = -4x^3 + 9x^2 + 20x \][/tex]
[tex]\[ f''(x) = \frac{d}{dx}(-4x^3 + 9x^2 + 20x) = -12x^2 + 18x + 20 \][/tex]
- Behavior at [tex]\( x = 0 \)[/tex]:
[tex]\[ f'(0) = -4(0)^3 + 9(0)^2 + 20(0) = 0 \][/tex]
[tex]\[ f''(0) = -12(0)^2 + 18(0) + 20 = 20 \quad (\neq 0) \][/tex]
Since [tex]\( f'(0) = 0 \)[/tex] and [tex]\( f''(0) \)[/tex] is not zero, the graph touches the [tex]\( x \)[/tex]-axis at [tex]\( x = 0 \)[/tex].
- Behavior at [tex]\( x = 5 \)[/tex]:
[tex]\[ f'(5) = -4(5)^3 + 9(5)^2 + 20(5) = -500 + 225 + 100 = -175 \quad (\neq 0) \][/tex]
Since [tex]\( f'(5) \neq 0 \)[/tex], the graph crosses the [tex]\( x \)[/tex]-axis at [tex]\( x = 5 \)[/tex].
- Behavior at [tex]\( x = -2 \)[/tex]:
[tex]\[ f'(-2) = -4(-2)^3 + 9(-2)^2 + 20(-2) = 32 + 36 - 40 = 28 \quad (\neq 0) \][/tex]
Since [tex]\( f'(-2) \neq 0 \)[/tex], the graph crosses the [tex]\( x \)[/tex]-axis at [tex]\( x = -2 \)[/tex].
Based on this analysis, the correct statement is:
The graph touches the [tex]\( x \)[/tex]-axis at [tex]\( x = 0 \)[/tex] and crosses the [tex]\( x \)[/tex]-axis at [tex]\( x = 5 \)[/tex] and [tex]\( x = -2 \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{2} \][/tex]
Here is the step-by-step process:
1. Find the [tex]\( x \)[/tex]-intercepts: Solve [tex]\( f(x) = 0 \)[/tex].
Given the function [tex]\( f(x) = -x^4 + 3x^3 + 10x^2 \)[/tex], we solve:
[tex]\[ -x^4 + 3x^3 + 10x^2 = 0 \][/tex]
Factor the equation:
[tex]\[ x^2(-x^2 + 3x + 10) = 0 \][/tex]
This gives us two factors:
[tex]\[ x^2 = 0 \quad \text{and} \quad -x^2 + 3x + 10 = 0 \][/tex]
Solving [tex]\( x^2 = 0 \)[/tex]:
[tex]\[ x = 0 \][/tex]
To solve [tex]\( -x^2 + 3x + 10 = 0 \)[/tex], we can use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = -1 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = 10 \)[/tex]:
[tex]\[ x = \frac{-3 \pm \sqrt{3^2 - 4(-1)(10)}}{2(-1)} = \frac{-3 \pm \sqrt{9 + 40}}{-2} = \frac{-3 \pm \sqrt{49}}{-2} = \frac{-3 \pm 7}{-2} \][/tex]
This gives:
[tex]\[ x = \frac{-3 + 7}{-2} = \frac{4}{-2} = -2 \][/tex]
and
[tex]\[ x = \frac{-3 - 7}{-2} = \frac{-10}{-2} = 5 \][/tex]
Thus, the roots are [tex]\( x = 0, x = 5, \)[/tex] and [tex]\( x = -2 \)[/tex].
2. Determine the behavior at each intercept: Check whether the graph touches or crosses the [tex]\( x \)[/tex]-axis at each intercept by analyzing the first and second derivatives at those points.
- First and Second Derivatives:
[tex]\[ f'(x) = \frac{d}{dx}(-x^4 + 3x^3 + 10x^2) = -4x^3 + 9x^2 + 20x \][/tex]
[tex]\[ f''(x) = \frac{d}{dx}(-4x^3 + 9x^2 + 20x) = -12x^2 + 18x + 20 \][/tex]
- Behavior at [tex]\( x = 0 \)[/tex]:
[tex]\[ f'(0) = -4(0)^3 + 9(0)^2 + 20(0) = 0 \][/tex]
[tex]\[ f''(0) = -12(0)^2 + 18(0) + 20 = 20 \quad (\neq 0) \][/tex]
Since [tex]\( f'(0) = 0 \)[/tex] and [tex]\( f''(0) \)[/tex] is not zero, the graph touches the [tex]\( x \)[/tex]-axis at [tex]\( x = 0 \)[/tex].
- Behavior at [tex]\( x = 5 \)[/tex]:
[tex]\[ f'(5) = -4(5)^3 + 9(5)^2 + 20(5) = -500 + 225 + 100 = -175 \quad (\neq 0) \][/tex]
Since [tex]\( f'(5) \neq 0 \)[/tex], the graph crosses the [tex]\( x \)[/tex]-axis at [tex]\( x = 5 \)[/tex].
- Behavior at [tex]\( x = -2 \)[/tex]:
[tex]\[ f'(-2) = -4(-2)^3 + 9(-2)^2 + 20(-2) = 32 + 36 - 40 = 28 \quad (\neq 0) \][/tex]
Since [tex]\( f'(-2) \neq 0 \)[/tex], the graph crosses the [tex]\( x \)[/tex]-axis at [tex]\( x = -2 \)[/tex].
Based on this analysis, the correct statement is:
The graph touches the [tex]\( x \)[/tex]-axis at [tex]\( x = 0 \)[/tex] and crosses the [tex]\( x \)[/tex]-axis at [tex]\( x = 5 \)[/tex] and [tex]\( x = -2 \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{2} \][/tex]
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