IDNLearn.com makes it easy to find answers and share knowledge with others. Join our community to receive timely and reliable responses to your questions from knowledgeable professionals.
Sagot :
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]
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Thank you for trusting IDNLearn.com with your questions. Visit us again for clear, concise, and accurate answers.
