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The function y = tan–1 x – x
- a)is always decreasing
- b)is always increasing
- c)first increases and then decreases
- d)first decreases and then increases
Correct answer is option 'A'. Can you explain this answer?
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The function y = tan1 x xa)is always decreasingb)is always increasing...
Understanding the Function y = tan(1/x)
The function \( y = \tan(1/x) \) exhibits specific behavior based on its mathematical properties.
1. Domain of the Function
- The function is defined for all \( x \) except \( x = 0 \).
- As \( x \) approaches 0 from either side, \( 1/x \) approaches infinity, causing \( \tan(1/x) \) to oscillate between positive and negative infinity.
2. Behavior of the Function
- For positive \( x \), \( 1/x \) decreases as \( x \) increases.
- The \( \tan \) function is periodic and increases from \( -\infty \) to \( +\infty \) within each interval of \( \pi \).
3. Derivative Analysis
- To determine if \( y = \tan(1/x) \) is increasing or decreasing, we can look at the derivative.
- The derivative \( y' = \sec^2(1/x) \cdot (-1/x^2) \) indicates that:
- \( \sec^2(1/x) \) is always positive where defined.
- The factor \( -1/x^2 \) is negative for all \( x \neq 0 \).
- Hence, \( y' < 0="" \)="" for="" all="" \(="" x="" /> 0 \) and \( x < 0="" \),="" confirming="" that="" the="" function="" is="" decreasing="" in="" both="" />
4. Conclusion
- Since the function consistently decreases and does not have any intervals of increase, the correct answer is option 'A': the function \( y = \tan(1/x) \) is always decreasing.
This analysis highlights the monotonic behavior of the function across its domain.
The function \( y = \tan(1/x) \) exhibits specific behavior based on its mathematical properties.
1. Domain of the Function
- The function is defined for all \( x \) except \( x = 0 \).
- As \( x \) approaches 0 from either side, \( 1/x \) approaches infinity, causing \( \tan(1/x) \) to oscillate between positive and negative infinity.
2. Behavior of the Function
- For positive \( x \), \( 1/x \) decreases as \( x \) increases.
- The \( \tan \) function is periodic and increases from \( -\infty \) to \( +\infty \) within each interval of \( \pi \).
3. Derivative Analysis
- To determine if \( y = \tan(1/x) \) is increasing or decreasing, we can look at the derivative.
- The derivative \( y' = \sec^2(1/x) \cdot (-1/x^2) \) indicates that:
- \( \sec^2(1/x) \) is always positive where defined.
- The factor \( -1/x^2 \) is negative for all \( x \neq 0 \).
- Hence, \( y' < 0="" \)="" for="" all="" \(="" x="" /> 0 \) and \( x < 0="" \),="" confirming="" that="" the="" function="" is="" decreasing="" in="" both="" />
4. Conclusion
- Since the function consistently decreases and does not have any intervals of increase, the correct answer is option 'A': the function \( y = \tan(1/x) \) is always decreasing.
This analysis highlights the monotonic behavior of the function across its domain.
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The function y = tan1 x xa)is always decreasingb)is always increasingc)first increases and then decreasesd)first decreases and then increasesCorrect answer is option 'A'. Can you explain this answer? for NDA 2025 is part of NDA preparation. The Question and answers have been prepared according to the NDA exam syllabus. Information about The function y = tan1 x xa)is always decreasingb)is always increasingc)first increases and then decreasesd)first decreases and then increasesCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for NDA 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The function y = tan1 x xa)is always decreasingb)is always increasingc)first increases and then decreasesd)first decreases and then increasesCorrect answer is option 'A'. Can you explain this answer?.
The function y = tan1 x xa)is always decreasingb)is always increasingc)first increases and then decreasesd)first decreases and then increasesCorrect answer is option 'A'. Can you explain this answer? for NDA 2025 is part of NDA preparation. The Question and answers have been prepared according to the NDA exam syllabus. Information about The function y = tan1 x xa)is always decreasingb)is always increasingc)first increases and then decreasesd)first decreases and then increasesCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for NDA 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The function y = tan1 x xa)is always decreasingb)is always increasingc)first increases and then decreasesd)first decreases and then increasesCorrect answer is option 'A'. Can you explain this answer?.
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