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Let f : [0, 10] → [0 ,10] be continuous function then
- a)f need not to have any fixed point
- b)f has atleast 10 fixed point
- c)f has atleast 9 fixed points
- d)f has atleast one fixed point
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
Let f : [0, 10] → [0 ,10] be continuous function thena)f need not...
Understanding Fixed Points
A fixed point of a function f is a point x such that f(x) = x. In the context of the problem, we analyze the continuous function f defined on the interval [0, 10].
Applying the Intermediate Value Theorem
Since f is continuous on a closed interval [0, 10], we can apply the Intermediate Value Theorem (IVT).
Evaluating f at the Endpoints
- Calculate f(0): Since f maps to [0, 10], f(0) is in the interval [0, 10].
- Calculate f(10): Similarly, f(10) is also in [0, 10].
Comparing Values
- If f(0) > 0, then f(0) cannot be a fixed point.
- If f(10) < 10,="" then="" f(10)="" cannot="" be="" a="" fixed="" />
Now, let's consider two scenarios:
1. If f(0) < /> This is impossible since f(0) is in [0, 10].
2. If f(0) > 0: Then there is a potential for a fixed point between [0, f(0)].
3. If f(10) < /> Then there is a potential for a fixed point between [f(10), 10].
Conclusion via the IVT
- At least one of the endpoints must lead to a scenario where f crosses the line y = x.
- Since f(0) and f(10) are both defined within the interval, and considering the continuity of f, the function must intersect the line y = x at least once.
Thus, by the continuity of f and the properties of the interval, we conclude:
Final Answer
There must be at least one fixed point in the interval [0, 10], confirming the correctness of option D.
A fixed point of a function f is a point x such that f(x) = x. In the context of the problem, we analyze the continuous function f defined on the interval [0, 10].
Applying the Intermediate Value Theorem
Since f is continuous on a closed interval [0, 10], we can apply the Intermediate Value Theorem (IVT).
Evaluating f at the Endpoints
- Calculate f(0): Since f maps to [0, 10], f(0) is in the interval [0, 10].
- Calculate f(10): Similarly, f(10) is also in [0, 10].
Comparing Values
- If f(0) > 0, then f(0) cannot be a fixed point.
- If f(10) < 10,="" then="" f(10)="" cannot="" be="" a="" fixed="" />
Now, let's consider two scenarios:
1. If f(0) < /> This is impossible since f(0) is in [0, 10].
2. If f(0) > 0: Then there is a potential for a fixed point between [0, f(0)].
3. If f(10) < /> Then there is a potential for a fixed point between [f(10), 10].
Conclusion via the IVT
- At least one of the endpoints must lead to a scenario where f crosses the line y = x.
- Since f(0) and f(10) are both defined within the interval, and considering the continuity of f, the function must intersect the line y = x at least once.
Thus, by the continuity of f and the properties of the interval, we conclude:
Final Answer
There must be at least one fixed point in the interval [0, 10], confirming the correctness of option D.
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Let f : [0, 10] → [0 ,10] be continuous function thena)f need not to have any fixed pointb)f has atleast 10 fixed pointc)f has atleast 9 fixed pointsd)f has atleast one fixed pointCorrect answer is option 'D'. Can you explain this answer?
Question Description
Let f : [0, 10] → [0 ,10] be continuous function thena)f need not to have any fixed pointb)f has atleast 10 fixed pointc)f has atleast 9 fixed pointsd)f has atleast one fixed pointCorrect answer is option 'D'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let f : [0, 10] → [0 ,10] be continuous function thena)f need not to have any fixed pointb)f has atleast 10 fixed pointc)f has atleast 9 fixed pointsd)f has atleast one fixed pointCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f : [0, 10] → [0 ,10] be continuous function thena)f need not to have any fixed pointb)f has atleast 10 fixed pointc)f has atleast 9 fixed pointsd)f has atleast one fixed pointCorrect answer is option 'D'. Can you explain this answer?.
Let f : [0, 10] → [0 ,10] be continuous function thena)f need not to have any fixed pointb)f has atleast 10 fixed pointc)f has atleast 9 fixed pointsd)f has atleast one fixed pointCorrect answer is option 'D'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let f : [0, 10] → [0 ,10] be continuous function thena)f need not to have any fixed pointb)f has atleast 10 fixed pointc)f has atleast 9 fixed pointsd)f has atleast one fixed pointCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f : [0, 10] → [0 ,10] be continuous function thena)f need not to have any fixed pointb)f has atleast 10 fixed pointc)f has atleast 9 fixed pointsd)f has atleast one fixed pointCorrect answer is option 'D'. Can you explain this answer?.
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