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Let f / [0, 1] -> R be a continuous function and assumes only rational values. Also f(0) = 1 and g(x) = f(1/2) * x ^ 2 - 4f(1/4) * x + 3f(2) If g(|x|) = k has four distinct roots, then range of 'k'is?
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Let f / [0, 1] -> R be a continuous function and assumes only rational...
Range of k in the given scenario
- Given Information:
- f(x) is a continuous function on [0, 1] and takes only rational values.
- f(0) = 1
- g(x) = f(1/2) * x^2 - 4f(1/4) * x + 3f(2)
- g(|x|) = k has four distinct roots
- Understanding the Problem:
- We need to find the range of k when g(|x|) = k has four distinct roots.
- As g(|x|) is a quadratic function, it will have at most two distinct roots unless its discriminant is positive.
- Key Points to Note:
- Since f(x) takes rational values and is continuous, f(x) must be constant in [0, 1].
- Given f(0) = 1, f(x) = 1 for all x in [0, 1].
- Substituting f(x) = 1 in g(x), we get g(x) = x^2 - 4x + 3 = (x - 1)(x - 3).
- The roots of g(x) are 1 and 3.
- Finding the Range of k:
- As g(|x|) has four distinct roots, the graph of g(x) must intersect the x-axis four times.
- Since g(x) = (x - 1)(x - 3), the roots of g(|x|) = k will be -3, -1, 1, and 3.
- Therefore, the range of k is (-3, 3).
By following the steps mentioned above and understanding the key points, we can determine that the range of k in the given scenario is (-3, 3).
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Let f / [0, 1] -> R be a continuous function and assumes only rational values. Also f(0) = 1 and g(x) = f(1/2) * x ^ 2 - 4f(1/4) * x + 3f(2) If g(|x|) = k has four distinct roots, then range of 'k'is?
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Let f / [0, 1] -> R be a continuous function and assumes only rational values. Also f(0) = 1 and g(x) = f(1/2) * x ^ 2 - 4f(1/4) * x + 3f(2) If g(|x|) = k has four distinct roots, then range of 'k'is? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let f / [0, 1] -> R be a continuous function and assumes only rational values. Also f(0) = 1 and g(x) = f(1/2) * x ^ 2 - 4f(1/4) * x + 3f(2) If g(|x|) = k has four distinct roots, then range of 'k'is? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f / [0, 1] -> R be a continuous function and assumes only rational values. Also f(0) = 1 and g(x) = f(1/2) * x ^ 2 - 4f(1/4) * x + 3f(2) If g(|x|) = k has four distinct roots, then range of 'k'is?.
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