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If for the twice differentiable function, f'(x) = f ''(x) = 0, x is
- a)a point of minima
- b)neither maxima nor minima
- c)can’t say
- d)a point of maxima
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
If for the twice differentiable function,f'(x) =f ''(x) = ...
If f'(x) = f''(x) = 0, then to tell whether x is a point of maxima or minima, we need to know the state of all its higher derivatives.
The correct answer is: can’t say
Most Upvoted Answer
If for the twice differentiable function,f'(x) =f ''(x) = ...
Understanding the Function f(x) = 0
When dealing with the function f(x) = 0, we need to analyze the implications of this equation in the context of minima and maxima.
1. Definition of Critical Points
- A critical point occurs where the first derivative f'(x) = 0 or is undefined.
- At this point, the function can either have a maximum, minimum, or be a point of inflection.
2. Second Derivative Test
- To determine the nature of the critical point, we can use the second derivative f''(x).
- If f''(x) > 0, the point is a local minimum.
- If f''(x) < 0,="" the="" point="" is="" a="" local="" />
- If f''(x) = 0, the test is inconclusive.
3. Analyzing f(x) = 0
- In this case, since f(x) = 0 for all x, we have f'(x) = 0 and f''(x) = 0 for any x.
- This means that the second derivative test is inconclusive because we cannot determine whether f(x) has a local minimum, local maximum, or neither.
4. Conclusion
- Since both the first and second derivatives are zero, we cannot categorically classify the point as a maximum or minimum.
- Therefore, the correct answer is option "C": we can't say whether it is a point of minima, maxima, or neither.
The nature of the function f(x) = 0 leads to an indeterminate situation, making it impossible to draw conclusions about its critical points without further information.
When dealing with the function f(x) = 0, we need to analyze the implications of this equation in the context of minima and maxima.
1. Definition of Critical Points
- A critical point occurs where the first derivative f'(x) = 0 or is undefined.
- At this point, the function can either have a maximum, minimum, or be a point of inflection.
2. Second Derivative Test
- To determine the nature of the critical point, we can use the second derivative f''(x).
- If f''(x) > 0, the point is a local minimum.
- If f''(x) < 0,="" the="" point="" is="" a="" local="" />
- If f''(x) = 0, the test is inconclusive.
3. Analyzing f(x) = 0
- In this case, since f(x) = 0 for all x, we have f'(x) = 0 and f''(x) = 0 for any x.
- This means that the second derivative test is inconclusive because we cannot determine whether f(x) has a local minimum, local maximum, or neither.
4. Conclusion
- Since both the first and second derivatives are zero, we cannot categorically classify the point as a maximum or minimum.
- Therefore, the correct answer is option "C": we can't say whether it is a point of minima, maxima, or neither.
The nature of the function f(x) = 0 leads to an indeterminate situation, making it impossible to draw conclusions about its critical points without further information.
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If for the twice differentiable function,f'(x) =f ''(x) = 0,xisa)a point of minimab)neither maxima nor minimac)can’t sayd)a point of maximaCorrect answer is option 'C'. Can you explain this answer?
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