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Let f : ℝ → ℝ be continuous on ℝ and differentiable on (−∞, 0) ∪ (0, ∞). Which of the following statements is (are) always TRUE?
- a)If is differentiable at 0 and f ′(0) = 0, then has a local maximum or a local minimum at 0
- b)If f has a local minimum at 0, then f is differentiable at 0 and f ′(0) = 0
- c)If f ′(x) < 0 for all x < 0 and f ′ (x) > 0 for all x > 0, then f has a global maximum at 0
- d)If f ′ (x) > 0 for all x < 0 and f ′(x) < 0 for all x > 0, then f has a global maximum at 0
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
Let f : ℝ → ℝ be continuous on ℝ and differenti...
Understanding the Problem
The function f is continuous everywhere and differentiable on (-∞, 0) ∪ (0, ∞). We need to analyze the given statements about the function at the point x = 0.
Statement Analysis
- Statement a: If f is differentiable at 0 and f'(0) = 0, then f has a local maximum or a local minimum at 0.
- This is not always true. A function can have f'(0) = 0 and still be neither a maximum nor a minimum (e.g., f(x) = x^3 at x = 0).
- Statement b: If f has a local minimum at 0, then f is differentiable at 0 and f'(0) = 0.
- This is also not necessarily true. A function can have a local minimum at a point without being differentiable there (e.g., f(x) = |x| has a minimum at x = 0 but is not differentiable at that point).
- Statement c: If f'(x) < 0="" for="" all="" x="" />< 0="" and="" f'(x)="" /> 0 for all x > 0, then f has a global maximum at 0.
- This statement is true under the condition of continuity, but it does not encompass all function behaviors. Thus, it is not universally valid.
- Statement d: If f'(x) > 0 for all x < 0="" and="" f'(x)="" />< 0="" for="" all="" x="" /> 0, then f has a global maximum at 0.
- This statement is true. If the derivative is positive to the left of 0 and negative to the right, it indicates that the function is increasing as it approaches 0 and then decreasing afterward. Therefore, f has a global maximum at x = 0.
Conclusion
The correct answer is option 'D' because the behavior of the derivative confirms that 0 is indeed a global maximum based on the criteria provided.
The function f is continuous everywhere and differentiable on (-∞, 0) ∪ (0, ∞). We need to analyze the given statements about the function at the point x = 0.
Statement Analysis
- Statement a: If f is differentiable at 0 and f'(0) = 0, then f has a local maximum or a local minimum at 0.
- This is not always true. A function can have f'(0) = 0 and still be neither a maximum nor a minimum (e.g., f(x) = x^3 at x = 0).
- Statement b: If f has a local minimum at 0, then f is differentiable at 0 and f'(0) = 0.
- This is also not necessarily true. A function can have a local minimum at a point without being differentiable there (e.g., f(x) = |x| has a minimum at x = 0 but is not differentiable at that point).
- Statement c: If f'(x) < 0="" for="" all="" x="" />< 0="" and="" f'(x)="" /> 0 for all x > 0, then f has a global maximum at 0.
- This statement is true under the condition of continuity, but it does not encompass all function behaviors. Thus, it is not universally valid.
- Statement d: If f'(x) > 0 for all x < 0="" and="" f'(x)="" />< 0="" for="" all="" x="" /> 0, then f has a global maximum at 0.
- This statement is true. If the derivative is positive to the left of 0 and negative to the right, it indicates that the function is increasing as it approaches 0 and then decreasing afterward. Therefore, f has a global maximum at x = 0.
Conclusion
The correct answer is option 'D' because the behavior of the derivative confirms that 0 is indeed a global maximum based on the criteria provided.
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Let f : ℝ → ℝ be continuous on ℝ and differentiable on (−∞, 0) ∪ (0, ∞). Which of the following statements is (are) always TRUE?a)If is differentiable at 0 and f′(0) = 0, then has a local maximum or a local minimum at 0b)If fhas a local minimum at 0, then fis differentiable at 0 and f′(0) = 0c)If f′(x) < 0 for all x< 0 and f′ (x) > 0 for all x> 0, then fhas a global maximum at 0d)If f′ (x) > 0 for all x< 0 and f′(x) < 0 for all x> 0, then fhas a global maximum at 0Correct answer is option 'D'. Can you explain this answer?
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Let f : ℝ → ℝ be continuous on ℝ and differentiable on (−∞, 0) ∪ (0, ∞). Which of the following statements is (are) always TRUE?a)If is differentiable at 0 and f′(0) = 0, then has a local maximum or a local minimum at 0b)If fhas a local minimum at 0, then fis differentiable at 0 and f′(0) = 0c)If f′(x) < 0 for all x< 0 and f′ (x) > 0 for all x> 0, then fhas a global maximum at 0d)If f′ (x) > 0 for all x< 0 and f′(x) < 0 for all x> 0, then fhas a global maximum at 0Correct answer is option 'D'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Let f : ℝ → ℝ be continuous on ℝ and differentiable on (−∞, 0) ∪ (0, ∞). Which of the following statements is (are) always TRUE?a)If is differentiable at 0 and f′(0) = 0, then has a local maximum or a local minimum at 0b)If fhas a local minimum at 0, then fis differentiable at 0 and f′(0) = 0c)If f′(x) < 0 for all x< 0 and f′ (x) > 0 for all x> 0, then fhas a global maximum at 0d)If f′ (x) > 0 for all x< 0 and f′(x) < 0 for all x> 0, then fhas a global maximum at 0Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f : ℝ → ℝ be continuous on ℝ and differentiable on (−∞, 0) ∪ (0, ∞). Which of the following statements is (are) always TRUE?a)If is differentiable at 0 and f′(0) = 0, then has a local maximum or a local minimum at 0b)If fhas a local minimum at 0, then fis differentiable at 0 and f′(0) = 0c)If f′(x) < 0 for all x< 0 and f′ (x) > 0 for all x> 0, then fhas a global maximum at 0d)If f′ (x) > 0 for all x< 0 and f′(x) < 0 for all x> 0, then fhas a global maximum at 0Correct answer is option 'D'. Can you explain this answer?.
Let f : ℝ → ℝ be continuous on ℝ and differentiable on (−∞, 0) ∪ (0, ∞). Which of the following statements is (are) always TRUE?a)If is differentiable at 0 and f′(0) = 0, then has a local maximum or a local minimum at 0b)If fhas a local minimum at 0, then fis differentiable at 0 and f′(0) = 0c)If f′(x) < 0 for all x< 0 and f′ (x) > 0 for all x> 0, then fhas a global maximum at 0d)If f′ (x) > 0 for all x< 0 and f′(x) < 0 for all x> 0, then fhas a global maximum at 0Correct answer is option 'D'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Let f : ℝ → ℝ be continuous on ℝ and differentiable on (−∞, 0) ∪ (0, ∞). Which of the following statements is (are) always TRUE?a)If is differentiable at 0 and f′(0) = 0, then has a local maximum or a local minimum at 0b)If fhas a local minimum at 0, then fis differentiable at 0 and f′(0) = 0c)If f′(x) < 0 for all x< 0 and f′ (x) > 0 for all x> 0, then fhas a global maximum at 0d)If f′ (x) > 0 for all x< 0 and f′(x) < 0 for all x> 0, then fhas a global maximum at 0Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f : ℝ → ℝ be continuous on ℝ and differentiable on (−∞, 0) ∪ (0, ∞). Which of the following statements is (are) always TRUE?a)If is differentiable at 0 and f′(0) = 0, then has a local maximum or a local minimum at 0b)If fhas a local minimum at 0, then fis differentiable at 0 and f′(0) = 0c)If f′(x) < 0 for all x< 0 and f′ (x) > 0 for all x> 0, then fhas a global maximum at 0d)If f′ (x) > 0 for all x< 0 and f′(x) < 0 for all x> 0, then fhas a global maximum at 0Correct answer is option 'D'. Can you explain this answer?.
Solutions for Let f : ℝ → ℝ be continuous on ℝ and differentiable on (−∞, 0) ∪ (0, ∞). Which of the following statements is (are) always TRUE?a)If is differentiable at 0 and f′(0) = 0, then has a local maximum or a local minimum at 0b)If fhas a local minimum at 0, then fis differentiable at 0 and f′(0) = 0c)If f′(x) < 0 for all x< 0 and f′ (x) > 0 for all x> 0, then fhas a global maximum at 0d)If f′ (x) > 0 for all x< 0 and f′(x) < 0 for all x> 0, then fhas a global maximum at 0Correct answer is option 'D'. Can you explain this answer? in English & in Hindi are available as part of our courses for IIT JAM.
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Let f : ℝ → ℝ be continuous on ℝ and differentiable on (−∞, 0) ∪ (0, ∞). Which of the following statements is (are) always TRUE?a)If is differentiable at 0 and f′(0) = 0, then has a local maximum or a local minimum at 0b)If fhas a local minimum at 0, then fis differentiable at 0 and f′(0) = 0c)If f′(x) < 0 for all x< 0 and f′ (x) > 0 for all x> 0, then fhas a global maximum at 0d)If f′ (x) > 0 for all x< 0 and f′(x) < 0 for all x> 0, then fhas a global maximum at 0Correct answer is option 'D'. Can you explain this answer?, a detailed solution for Let f : ℝ → ℝ be continuous on ℝ and differentiable on (−∞, 0) ∪ (0, ∞). Which of the following statements is (are) always TRUE?a)If is differentiable at 0 and f′(0) = 0, then has a local maximum or a local minimum at 0b)If fhas a local minimum at 0, then fis differentiable at 0 and f′(0) = 0c)If f′(x) < 0 for all x< 0 and f′ (x) > 0 for all x> 0, then fhas a global maximum at 0d)If f′ (x) > 0 for all x< 0 and f′(x) < 0 for all x> 0, then fhas a global maximum at 0Correct answer is option 'D'. Can you explain this answer? has been provided alongside types of Let f : ℝ → ℝ be continuous on ℝ and differentiable on (−∞, 0) ∪ (0, ∞). Which of the following statements is (are) always TRUE?a)If is differentiable at 0 and f′(0) = 0, then has a local maximum or a local minimum at 0b)If fhas a local minimum at 0, then fis differentiable at 0 and f′(0) = 0c)If f′(x) < 0 for all x< 0 and f′ (x) > 0 for all x> 0, then fhas a global maximum at 0d)If f′ (x) > 0 for all x< 0 and f′(x) < 0 for all x> 0, then fhas a global maximum at 0Correct answer is option 'D'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Let f : ℝ → ℝ be continuous on ℝ and differentiable on (−∞, 0) ∪ (0, ∞). Which of the following statements is (are) always TRUE?a)If is differentiable at 0 and f′(0) = 0, then has a local maximum or a local minimum at 0b)If fhas a local minimum at 0, then fis differentiable at 0 and f′(0) = 0c)If f′(x) < 0 for all x< 0 and f′ (x) > 0 for all x> 0, then fhas a global maximum at 0d)If f′ (x) > 0 for all x< 0 and f′(x) < 0 for all x> 0, then fhas a global maximum at 0Correct answer is option 'D'. Can you explain this answer? tests, examples and also practice IIT JAM tests.
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