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Let g (x) be continuous in a neighbourhood of ‘a’ and g (a) ≠ 0. Let f be a function such that f ‘ (x) = g(x) (x−a)2 , then
- a)f is decreasing at a if g (a) >
- b)f is increasing at a if g (a) < 0
- c)f is increasing at a if g (a) > 0
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
Let g (x) be continuous in a neighbourhood of ‘a’ and g (a...
Since g is continuous at a , therefore , if g (a) > 0 , then there is a nhd.of a, say (a-e , a+ e) in which g (x) is positive .This means that f ‘ (x)>0 in this nhd of a and hence f (x) is increasing at a.
Most Upvoted Answer
Let g (x) be continuous in a neighbourhood of ‘a’ and g (a...
Understanding the Function f(x)
To analyze the behavior of the function f(x) = g(x)(x - a)² around a, we first need to consider the components involved.
Behavior of g(x)
- g(x) is continuous in a neighborhood of a.
- g(a) is not equal to 0, which means g(a) could be positive or negative.
Analyzing f(x)
The function f(x) can be expressed as the product of g(x) and (x - a)².
- The term (x - a)² is always non-negative and equals zero when x = a.
- This means that around x = a, f(x) will depend on the behavior of g(x).
Finding the Derivative f'(x)
To determine if f(x) is increasing or decreasing at a, we can find the derivative f'(x).
- Using the product rule, we have:
f'(x) = g'(x)(x - a)² + g(x) * 2(x - a).
At x = a:
- f'(a) = g(a) * 0 + g(a) * 2 * 0 = 0.
This indicates that we need to analyze further around a.
Sign of g(a)
Now, we consider the signs of g(a):
- If g(a) > 0:
- As x approaches a, f(x) will be greater than zero for values close to a, indicating f(x) is increasing.
- If g(a) < />
- The behavior of f(x) depends on the continuity of g(x). However, since we are focusing on the case where g(a) is positive, we conclude that f(x) is indeed increasing at a.
Conclusion
Thus, the correct answer is option C: f is increasing at a if g(a) > 0. This highlights the importance of the function g(x) in determining the behavior of f(x) around the point a.
To analyze the behavior of the function f(x) = g(x)(x - a)² around a, we first need to consider the components involved.
Behavior of g(x)
- g(x) is continuous in a neighborhood of a.
- g(a) is not equal to 0, which means g(a) could be positive or negative.
Analyzing f(x)
The function f(x) can be expressed as the product of g(x) and (x - a)².
- The term (x - a)² is always non-negative and equals zero when x = a.
- This means that around x = a, f(x) will depend on the behavior of g(x).
Finding the Derivative f'(x)
To determine if f(x) is increasing or decreasing at a, we can find the derivative f'(x).
- Using the product rule, we have:
f'(x) = g'(x)(x - a)² + g(x) * 2(x - a).
At x = a:
- f'(a) = g(a) * 0 + g(a) * 2 * 0 = 0.
This indicates that we need to analyze further around a.
Sign of g(a)
Now, we consider the signs of g(a):
- If g(a) > 0:
- As x approaches a, f(x) will be greater than zero for values close to a, indicating f(x) is increasing.
- If g(a) < />
- The behavior of f(x) depends on the continuity of g(x). However, since we are focusing on the case where g(a) is positive, we conclude that f(x) is indeed increasing at a.
Conclusion
Thus, the correct answer is option C: f is increasing at a if g(a) > 0. This highlights the importance of the function g(x) in determining the behavior of f(x) around the point a.
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Let g (x) be continuous in a neighbourhood of ‘a’ and g (a) ≠ 0. Let f be a function such that f ‘ (x) = g(x)(x−a)2 , thena)f is decreasing at a if g (a) >b)f is increasing at a if g (a) < 0c)f is increasing at a if g (a) > 0d)none of theseCorrect answer is option 'C'. Can you explain this answer?
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Let g (x) be continuous in a neighbourhood of ‘a’ and g (a) ≠ 0. Let f be a function such that f ‘ (x) = g(x)(x−a)2 , thena)f is decreasing at a if g (a) >b)f is increasing at a if g (a) < 0c)f is increasing at a if g (a) > 0d)none of theseCorrect answer is option 'C'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let g (x) be continuous in a neighbourhood of ‘a’ and g (a) ≠ 0. Let f be a function such that f ‘ (x) = g(x)(x−a)2 , thena)f is decreasing at a if g (a) >b)f is increasing at a if g (a) < 0c)f is increasing at a if g (a) > 0d)none of theseCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let g (x) be continuous in a neighbourhood of ‘a’ and g (a) ≠ 0. Let f be a function such that f ‘ (x) = g(x)(x−a)2 , thena)f is decreasing at a if g (a) >b)f is increasing at a if g (a) < 0c)f is increasing at a if g (a) > 0d)none of theseCorrect answer is option 'C'. Can you explain this answer?.
Let g (x) be continuous in a neighbourhood of ‘a’ and g (a) ≠ 0. Let f be a function such that f ‘ (x) = g(x)(x−a)2 , thena)f is decreasing at a if g (a) >b)f is increasing at a if g (a) < 0c)f is increasing at a if g (a) > 0d)none of theseCorrect answer is option 'C'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let g (x) be continuous in a neighbourhood of ‘a’ and g (a) ≠ 0. Let f be a function such that f ‘ (x) = g(x)(x−a)2 , thena)f is decreasing at a if g (a) >b)f is increasing at a if g (a) < 0c)f is increasing at a if g (a) > 0d)none of theseCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let g (x) be continuous in a neighbourhood of ‘a’ and g (a) ≠ 0. Let f be a function such that f ‘ (x) = g(x)(x−a)2 , thena)f is decreasing at a if g (a) >b)f is increasing at a if g (a) < 0c)f is increasing at a if g (a) > 0d)none of theseCorrect answer is option 'C'. Can you explain this answer?.
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