Explore a wide range of topics and get answers from experts on IDNLearn.com. Discover reliable answers to your questions with our extensive database of expert knowledge.
Sagot :
To find the [tex]\( x \)[/tex]-intercepts of the polynomial function [tex]\( f(x) = x^4 - 25x^2 \)[/tex], we need to solve the equation [tex]\( f(x) = 0 \)[/tex].
Step-by-step:
### Step 1: Factorize the polynomial
First, we can factorize [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = x^4 - 25x^2 \][/tex]
We notice that this can be written as a difference of squares:
[tex]\[ f(x) = (x^2)^2 - (5x)^2 \][/tex]
[tex]\[ f(x) = (x^2 - 5)(x^2 + 5) \][/tex]
### Step 2: Solve for [tex]\( x \)[/tex]
Set each factor equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x^2 - 5 = 0 \quad \text{and} \quad x^2 + 5 = 0 \][/tex]
Solve [tex]\( x^2 - 5 = 0 \)[/tex]:
[tex]\[ x^2 = 5 \][/tex]
[tex]\[ x = \pm \sqrt{5} \][/tex]
[tex]\[ x = \pm 5 \][/tex]
Solve [tex]\( x^2 + 5 = 0 \)[/tex]:
[tex]\[ x^2 = -5 \][/tex]
Since [tex]\( x^2 = -5 \)[/tex] has no real solutions, we ignore this part.
Thus, the [tex]\( x \)[/tex]-intercepts are:
[tex]\[ x = -5, 0, 5 \][/tex]
### Step 3: Determine the behavior at each intercept
We need to determine whether the graph crosses the [tex]\( x \)[/tex]-axis or touches the [tex]\( x \)[/tex]-axis and turns around at each intercept.
1. At [tex]\( x = -5 \)[/tex]:
- To determine the behavior, we use the second derivative test. If the second derivative at [tex]\( x = -5 \)[/tex] is positive, the graph touches the [tex]\( x \)[/tex]-axis and turns around; if negative, the graph crosses the [tex]\( x \)[/tex]-axis.
- The second derivative of [tex]\( f(x) \)[/tex] is [tex]\( 12x^2 - 50 \)[/tex].
- Evaluating the second derivative at [tex]\( x = -5 \)[/tex]:
[tex]\[ f''(-5) = 12 \times (-5)^2 - 50 = 12 \times 25 - 50 = 300 - 50 = 250 \][/tex]
- Since [tex]\( f''(-5) > 0 \)[/tex], the graph touches the [tex]\( x \)[/tex]-axis and turns around.
2. At [tex]\( x = 0 \)[/tex]:
- Evaluate the second derivative at [tex]\( x = 0 \)[/tex]:
[tex]\[ f''(0) = 12 \times (0)^2 - 50 = -50 \][/tex]
- Since [tex]\( f''(0) < 0 \)[/tex], the graph crosses the [tex]\( x \)[/tex]-axis.
3. At [tex]\( x = 5 \)[/tex]:
- Evaluate the second derivative at [tex]\( x = 5 \)[/tex]:
[tex]\[ f''(5) = 12 \times (5)^2 - 50 = 12 \times 25 - 50 = 300 - 50 = 250 \][/tex]
- Since [tex]\( f''(5) > 0 \)[/tex], the graph touches the [tex]\( x \)[/tex]-axis and turns around.
### Step 4: Conclusion
Putting it all together, the results are:
- At [tex]\( x = -5 \)[/tex]: touches the [tex]\( x \)[/tex]-axis and turns around.
- At [tex]\( x = 0 \)[/tex]: crosses the [tex]\( x \)[/tex]-axis.
- At [tex]\( x = 5 \)[/tex]: touches the [tex]\( x \)[/tex]-axis and turns around.
Based on the given choices, the correct answer is:
[tex]\[ \textbf{None of the provided options are correct, but the closest would be: } \][/tex]
[tex]\[ \text{0 , touches the } x \text{-axis and turns around; } 5 \text{ , touches the } x \text{-axis and turns around; } -5 \text{ , touches the } x \text{-axis and turns around} \][/tex]
Step-by-step:
### Step 1: Factorize the polynomial
First, we can factorize [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = x^4 - 25x^2 \][/tex]
We notice that this can be written as a difference of squares:
[tex]\[ f(x) = (x^2)^2 - (5x)^2 \][/tex]
[tex]\[ f(x) = (x^2 - 5)(x^2 + 5) \][/tex]
### Step 2: Solve for [tex]\( x \)[/tex]
Set each factor equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x^2 - 5 = 0 \quad \text{and} \quad x^2 + 5 = 0 \][/tex]
Solve [tex]\( x^2 - 5 = 0 \)[/tex]:
[tex]\[ x^2 = 5 \][/tex]
[tex]\[ x = \pm \sqrt{5} \][/tex]
[tex]\[ x = \pm 5 \][/tex]
Solve [tex]\( x^2 + 5 = 0 \)[/tex]:
[tex]\[ x^2 = -5 \][/tex]
Since [tex]\( x^2 = -5 \)[/tex] has no real solutions, we ignore this part.
Thus, the [tex]\( x \)[/tex]-intercepts are:
[tex]\[ x = -5, 0, 5 \][/tex]
### Step 3: Determine the behavior at each intercept
We need to determine whether the graph crosses the [tex]\( x \)[/tex]-axis or touches the [tex]\( x \)[/tex]-axis and turns around at each intercept.
1. At [tex]\( x = -5 \)[/tex]:
- To determine the behavior, we use the second derivative test. If the second derivative at [tex]\( x = -5 \)[/tex] is positive, the graph touches the [tex]\( x \)[/tex]-axis and turns around; if negative, the graph crosses the [tex]\( x \)[/tex]-axis.
- The second derivative of [tex]\( f(x) \)[/tex] is [tex]\( 12x^2 - 50 \)[/tex].
- Evaluating the second derivative at [tex]\( x = -5 \)[/tex]:
[tex]\[ f''(-5) = 12 \times (-5)^2 - 50 = 12 \times 25 - 50 = 300 - 50 = 250 \][/tex]
- Since [tex]\( f''(-5) > 0 \)[/tex], the graph touches the [tex]\( x \)[/tex]-axis and turns around.
2. At [tex]\( x = 0 \)[/tex]:
- Evaluate the second derivative at [tex]\( x = 0 \)[/tex]:
[tex]\[ f''(0) = 12 \times (0)^2 - 50 = -50 \][/tex]
- Since [tex]\( f''(0) < 0 \)[/tex], the graph crosses the [tex]\( x \)[/tex]-axis.
3. At [tex]\( x = 5 \)[/tex]:
- Evaluate the second derivative at [tex]\( x = 5 \)[/tex]:
[tex]\[ f''(5) = 12 \times (5)^2 - 50 = 12 \times 25 - 50 = 300 - 50 = 250 \][/tex]
- Since [tex]\( f''(5) > 0 \)[/tex], the graph touches the [tex]\( x \)[/tex]-axis and turns around.
### Step 4: Conclusion
Putting it all together, the results are:
- At [tex]\( x = -5 \)[/tex]: touches the [tex]\( x \)[/tex]-axis and turns around.
- At [tex]\( x = 0 \)[/tex]: crosses the [tex]\( x \)[/tex]-axis.
- At [tex]\( x = 5 \)[/tex]: touches the [tex]\( x \)[/tex]-axis and turns around.
Based on the given choices, the correct answer is:
[tex]\[ \textbf{None of the provided options are correct, but the closest would be: } \][/tex]
[tex]\[ \text{0 , touches the } x \text{-axis and turns around; } 5 \text{ , touches the } x \text{-axis and turns around; } -5 \text{ , touches the } x \text{-axis and turns around} \][/tex]
Thank you for using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. Your questions deserve reliable answers. Thanks for visiting IDNLearn.com, and see you again soon for more helpful information.
