Year 12 Exam > Year 12 Questions > Let f(x) = (x –4) (x –5) (x &ndas...
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Let f(x) = (x – 4) (x – 5) (x – 6) (x – 7) then,
- a)f '(x) = 0 has four roots
- b)Three roots of f'(x) = 0 lie in (4, 5) ∪ (5, 6) ∪ (6, 7)
- c)The equation f'(x) = 0 has only one real root
- d)Three roots of f'(x) = 0 lie in (3, 4) ∪ (4, 5) ∪ (5, 6)
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
Let f(x) = (x –4) (x –5) (x –6) (x –7) then,a)...
Explanation:
Given Function:
f(x) = (x - 4)(x - 5)(x - 6)(x - 7)
Number of Roots:
a) The given function f(x) is a polynomial of degree 4, which means it can have at most 4 roots.
Therefore, f(x) = 0 has four roots.
Roots in Intervals:
b) To find the roots in the intervals (4, 5), (5, 6), and (6, 7), we need to analyze the behavior of the function in these intervals.
- In the interval (4, 5):
- f(4) = (4 - 4)(4 - 5)(4 - 6)(4 - 7) = 0
- In the interval (5, 6):
- f(5) = (5 - 4)(5 - 5)(5 - 6)(5 - 7) = 0
- In the interval (6, 7):
- f(6) = (6 - 4)(6 - 5)(6 - 6)(6 - 7) = 0
Therefore, three roots of f(x) = 0 lie in the intervals (4, 5), (5, 6), and (6, 7).
Real Roots:
c) The given function is a polynomial of degree 4, so it can have up to 4 real roots.
Therefore, the equation f(x) = 0 can have more than one real root.
Roots in Other Intervals:
d) The intervals (3, 4), (4, 5), and (5, 6) also contain roots of the function f(x) = 0, as the function changes sign in these intervals. However, the question specifically asks about the intervals (4, 5), (5, 6), and (6, 7).
Therefore, the correct answer is option 'b', as three roots of f(x) = 0 lie in the intervals (4, 5), (5, 6), and (6, 7).
Given Function:
f(x) = (x - 4)(x - 5)(x - 6)(x - 7)
Number of Roots:
a) The given function f(x) is a polynomial of degree 4, which means it can have at most 4 roots.
Therefore, f(x) = 0 has four roots.
Roots in Intervals:
b) To find the roots in the intervals (4, 5), (5, 6), and (6, 7), we need to analyze the behavior of the function in these intervals.
- In the interval (4, 5):
- f(4) = (4 - 4)(4 - 5)(4 - 6)(4 - 7) = 0
- In the interval (5, 6):
- f(5) = (5 - 4)(5 - 5)(5 - 6)(5 - 7) = 0
- In the interval (6, 7):
- f(6) = (6 - 4)(6 - 5)(6 - 6)(6 - 7) = 0
Therefore, three roots of f(x) = 0 lie in the intervals (4, 5), (5, 6), and (6, 7).
Real Roots:
c) The given function is a polynomial of degree 4, so it can have up to 4 real roots.
Therefore, the equation f(x) = 0 can have more than one real root.
Roots in Other Intervals:
d) The intervals (3, 4), (4, 5), and (5, 6) also contain roots of the function f(x) = 0, as the function changes sign in these intervals. However, the question specifically asks about the intervals (4, 5), (5, 6), and (6, 7).
Therefore, the correct answer is option 'b', as three roots of f(x) = 0 lie in the intervals (4, 5), (5, 6), and (6, 7).
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Let f(x) = (x –4) (x –5) (x –6) (x –7) then,a)f (x) = 0 has four rootsb)Three roots of f(x) = 0 lie in (4, 5) ∪ (5, 6) ∪ (6, 7)c)The equation f(x) = 0 has only one real rootd)Three roots of f(x) = 0 lie in (3, 4) ∪ (4, 5) ∪ (5, 6)Correct answer is option 'B'. Can you explain this answer?
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Let f(x) = (x –4) (x –5) (x –6) (x –7) then,a)f (x) = 0 has four rootsb)Three roots of f(x) = 0 lie in (4, 5) ∪ (5, 6) ∪ (6, 7)c)The equation f(x) = 0 has only one real rootd)Three roots of f(x) = 0 lie in (3, 4) ∪ (4, 5) ∪ (5, 6)Correct answer is option 'B'. Can you explain this answer? for Year 12 2025 is part of Year 12 preparation. The Question and answers have been prepared according to the Year 12 exam syllabus. Information about Let f(x) = (x –4) (x –5) (x –6) (x –7) then,a)f (x) = 0 has four rootsb)Three roots of f(x) = 0 lie in (4, 5) ∪ (5, 6) ∪ (6, 7)c)The equation f(x) = 0 has only one real rootd)Three roots of f(x) = 0 lie in (3, 4) ∪ (4, 5) ∪ (5, 6)Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Year 12 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f(x) = (x –4) (x –5) (x –6) (x –7) then,a)f (x) = 0 has four rootsb)Three roots of f(x) = 0 lie in (4, 5) ∪ (5, 6) ∪ (6, 7)c)The equation f(x) = 0 has only one real rootd)Three roots of f(x) = 0 lie in (3, 4) ∪ (4, 5) ∪ (5, 6)Correct answer is option 'B'. Can you explain this answer?.
Let f(x) = (x –4) (x –5) (x –6) (x –7) then,a)f (x) = 0 has four rootsb)Three roots of f(x) = 0 lie in (4, 5) ∪ (5, 6) ∪ (6, 7)c)The equation f(x) = 0 has only one real rootd)Three roots of f(x) = 0 lie in (3, 4) ∪ (4, 5) ∪ (5, 6)Correct answer is option 'B'. Can you explain this answer? for Year 12 2025 is part of Year 12 preparation. The Question and answers have been prepared according to the Year 12 exam syllabus. Information about Let f(x) = (x –4) (x –5) (x –6) (x –7) then,a)f (x) = 0 has four rootsb)Three roots of f(x) = 0 lie in (4, 5) ∪ (5, 6) ∪ (6, 7)c)The equation f(x) = 0 has only one real rootd)Three roots of f(x) = 0 lie in (3, 4) ∪ (4, 5) ∪ (5, 6)Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Year 12 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f(x) = (x –4) (x –5) (x –6) (x –7) then,a)f (x) = 0 has four rootsb)Three roots of f(x) = 0 lie in (4, 5) ∪ (5, 6) ∪ (6, 7)c)The equation f(x) = 0 has only one real rootd)Three roots of f(x) = 0 lie in (3, 4) ∪ (4, 5) ∪ (5, 6)Correct answer is option 'B'. Can you explain this answer?.
Solutions for Let f(x) = (x –4) (x –5) (x –6) (x –7) then,a)f (x) = 0 has four rootsb)Three roots of f(x) = 0 lie in (4, 5) ∪ (5, 6) ∪ (6, 7)c)The equation f(x) = 0 has only one real rootd)Three roots of f(x) = 0 lie in (3, 4) ∪ (4, 5) ∪ (5, 6)Correct answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for Year 12.
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Let f(x) = (x –4) (x –5) (x –6) (x –7) then,a)f (x) = 0 has four rootsb)Three roots of f(x) = 0 lie in (4, 5) ∪ (5, 6) ∪ (6, 7)c)The equation f(x) = 0 has only one real rootd)Three roots of f(x) = 0 lie in (3, 4) ∪ (4, 5) ∪ (5, 6)Correct answer is option 'B'. Can you explain this answer?, a detailed solution for Let f(x) = (x –4) (x –5) (x –6) (x –7) then,a)f (x) = 0 has four rootsb)Three roots of f(x) = 0 lie in (4, 5) ∪ (5, 6) ∪ (6, 7)c)The equation f(x) = 0 has only one real rootd)Three roots of f(x) = 0 lie in (3, 4) ∪ (4, 5) ∪ (5, 6)Correct answer is option 'B'. Can you explain this answer? has been provided alongside types of Let f(x) = (x –4) (x –5) (x –6) (x –7) then,a)f (x) = 0 has four rootsb)Three roots of f(x) = 0 lie in (4, 5) ∪ (5, 6) ∪ (6, 7)c)The equation f(x) = 0 has only one real rootd)Three roots of f(x) = 0 lie in (3, 4) ∪ (4, 5) ∪ (5, 6)Correct answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Let f(x) = (x –4) (x –5) (x –6) (x –7) then,a)f (x) = 0 has four rootsb)Three roots of f(x) = 0 lie in (4, 5) ∪ (5, 6) ∪ (6, 7)c)The equation f(x) = 0 has only one real rootd)Three roots of f(x) = 0 lie in (3, 4) ∪ (4, 5) ∪ (5, 6)Correct answer is option 'B'. Can you explain this answer? tests, examples and also practice Year 12 tests.
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