Question
Use the Intermediate Value Theorem to choose an interval over which the function, f left parenthesis x right parenthesis equals negative 2 x cubed minus 3 x plus 5, is guaranteed to have a zero.
(1 point)
Responses
left bracket negative 3 comma negative 2 right bracket
Image with alt text: left bracket negative 3 comma negative 2 right bracket
left bracket negative 2 comma 0 right bracket
Image with alt text: left bracket negative 2 comma 0 right bracket
left bracket 0 comma 2 right bracket
Image with alt text: left bracket 0 comma 2 right bracket
left bracket 2 comma 4 right bracket
(1 point)
Responses
left bracket negative 3 comma negative 2 right bracket
Image with alt text: left bracket negative 3 comma negative 2 right bracket
left bracket negative 2 comma 0 right bracket
Image with alt text: left bracket negative 2 comma 0 right bracket
left bracket 0 comma 2 right bracket
Image with alt text: left bracket 0 comma 2 right bracket
left bracket 2 comma 4 right bracket
Answers
GPT-4o mini
To apply the Intermediate Value Theorem, we first need to evaluate the function \( f(x) = -2x^3 - 3x + 5 \) at the endpoints of the given intervals to see where the function changes sign. A change in sign indicates that there is at least one zero in that interval.
### 1. Check the interval \([-3, -2]\):
- \( f(-3) = -2(-3)^3 - 3(-3) + 5 = -2(-27) + 9 + 5 = 54 + 9 + 5 = 68 \)
- \( f(-2) = -2(-2)^3 - 3(-2) + 5 = -2(-8) + 6 + 5 = 16 + 6 + 5 = 27 \)
Both values are positive, so no zero in \([-3, -2]\).
### 2. Check the interval \([-2, 0]\):
- \( f(-2) = 27 \) (as calculated above)
- \( f(0) = -2(0)^3 - 3(0) + 5 = 5 \)
Both values are positive, so no zero in \([-2, 0]\).
### 3. Check the interval \([0, 2]\):
- \( f(0) = 5 \) (as calculated above)
- \( f(2) = -2(2)^3 - 3(2) + 5 = -2(8) - 6 + 5 = -16 - 6 + 5 = -17 \)
Here, \( f(0) = 5 \) and \( f(2) = -17 \) shows a change in sign (from positive to negative), indicating that there is at least one zero in the interval \([0, 2]\).
### 4. Check the interval \([2, 4]\):
- \( f(2) = -17 \) (as calculated above)
- \( f(4) = -2(4)^3 - 3(4) + 5 = -2(64) - 12 + 5 = -128 - 12 + 5 = -135 \)
Both values are negative, so no zero in \([2, 4]\).
### Conclusion:
The only interval where the function changes sign is \([0, 2]\). Thus, according to the Intermediate Value Theorem, the function \( f(x) \) is guaranteed to have a zero in the interval \([0, 2]\).
**Answer: \([0, 2]\)**
### 1. Check the interval \([-3, -2]\):
- \( f(-3) = -2(-3)^3 - 3(-3) + 5 = -2(-27) + 9 + 5 = 54 + 9 + 5 = 68 \)
- \( f(-2) = -2(-2)^3 - 3(-2) + 5 = -2(-8) + 6 + 5 = 16 + 6 + 5 = 27 \)
Both values are positive, so no zero in \([-3, -2]\).
### 2. Check the interval \([-2, 0]\):
- \( f(-2) = 27 \) (as calculated above)
- \( f(0) = -2(0)^3 - 3(0) + 5 = 5 \)
Both values are positive, so no zero in \([-2, 0]\).
### 3. Check the interval \([0, 2]\):
- \( f(0) = 5 \) (as calculated above)
- \( f(2) = -2(2)^3 - 3(2) + 5 = -2(8) - 6 + 5 = -16 - 6 + 5 = -17 \)
Here, \( f(0) = 5 \) and \( f(2) = -17 \) shows a change in sign (from positive to negative), indicating that there is at least one zero in the interval \([0, 2]\).
### 4. Check the interval \([2, 4]\):
- \( f(2) = -17 \) (as calculated above)
- \( f(4) = -2(4)^3 - 3(4) + 5 = -2(64) - 12 + 5 = -128 - 12 + 5 = -135 \)
Both values are negative, so no zero in \([2, 4]\).
### Conclusion:
The only interval where the function changes sign is \([0, 2]\). Thus, according to the Intermediate Value Theorem, the function \( f(x) \) is guaranteed to have a zero in the interval \([0, 2]\).
**Answer: \([0, 2]\)**