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If f is a real-valued differentiable function such that f(x) f’(x) < 0 for all real x, then
  • a)
     f(x) must be an increasing function
  • b)
     f(x) must be a decreasing function
  • c)
    |f (x )| must be an increasing function
  • d)
    |f (x)| must be a drecreasing function
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
If f is a real-valued differentiable function such that f(x) f’(...
 f(x) f (x) < 0  ....(i)
f(x) < 0 ; f (x) > 0   or f(x) > 0 ; f (x) < 0      ............... (ii)
So possible graphs of f(x) are
|f(x)| is decreasing function in both cases.
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Most Upvoted Answer
If f is a real-valued differentiable function such that f(x) f’(...
Explanation:
- Given Conditions:
Given that f(x) f'(x) < 0="" for="" all="" real="" />
This means that the product of the function f(x) and its derivative f'(x) is always negative.
- Understanding the Sign of f'(x):
Since the product f(x) f'(x) is negative for all x, this implies that f(x) and f'(x) have opposite signs.
If f'(x) is positive, then f(x) must be negative.
If f'(x) is negative, then f(x) must be positive.
- Absoluteness of f(x):
Since f(x) and f'(x) have opposite signs, the absolute value of f(x) will always be positive.
Therefore, |f(x)| is always positive regardless of the sign of f(x).
- Conclusion:
From the above analysis, we can conclude that |f(x)| must be a decreasing function.
This is because the absolute value of f(x) is always positive, and since f(x) and f'(x) have opposite signs, the magnitude of f(x) will decrease as x increases.
Therefore, the correct answer is option 'D' - |f(x)| must be a decreasing function.
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If f is a real-valued differentiable function such that f(x) f’(x) < 0 for all real x, thena)f(x) must be an increasing functionb)f(x) must be a decreasing functionc)|f (x )| must be an increasing functiond)|f (x)| must be a drecreasing functionCorrect answer is option 'D'. Can you explain this answer?
Question Description
If f is a real-valued differentiable function such that f(x) f’(x) < 0 for all real x, thena)f(x) must be an increasing functionb)f(x) must be a decreasing functionc)|f (x )| must be an increasing functiond)|f (x)| must be a drecreasing functionCorrect answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If f is a real-valued differentiable function such that f(x) f’(x) < 0 for all real x, thena)f(x) must be an increasing functionb)f(x) must be a decreasing functionc)|f (x )| must be an increasing functiond)|f (x)| must be a drecreasing functionCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If f is a real-valued differentiable function such that f(x) f’(x) < 0 for all real x, thena)f(x) must be an increasing functionb)f(x) must be a decreasing functionc)|f (x )| must be an increasing functiond)|f (x)| must be a drecreasing functionCorrect answer is option 'D'. Can you explain this answer?.
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