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Consider the polynomial f[x) = x3 - 6x2 + 11x - 6 on the domain S, given by 1 ≤ x ≤ 3. The first and second derivatives are f(x) and f'{x).
Consider the following statements:
I. The given polynomial is zero at the boundary points x = 1 and x = 3.
II. There exists one local maxima of f{x) within the domain S.
III. The second derivative f"(x) > 0 throughout the domains S.
IV. There exists one local minima f(x) within the domain S.
  • a)
    Only statements II and IV are correct.
  • b)
    Only statements I and IV are correct.
  • c)
    Only statements I. II and III are correct.
  • d)
    Only statements I. II and IV are correct.
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
Consider the polynomial f[x) = x3 - 6x2 + 11x - 6 on the domain S, giv...
Understanding the Polynomial Function
The given polynomial is f(x) = x³ - 6x² + 11x - 6 defined on the domain S: 1 ≤ x ≤ 3.
Statement I: Boundary Values
- The polynomial evaluates to zero at the boundary points:
- f(1) = 1 - 6 + 11 - 6 = 0
- f(3) = 27 - 54 + 33 - 6 = 0
- Therefore, Statement I is correct.
Statement II: Local Maxima
- To find local maxima, we first compute the first derivative:
- f'(x) = 3x² - 12x + 11
- Setting f'(x) = 0 to find critical points:
- The roots of the quadratic can be calculated using the quadratic formula, yielding one local maximum within the interval.
- Thus, Statement II is correct.
Statement III: Second Derivative Test
- The second derivative is:
- f''(x) = 6x - 12
- Evaluating f''(x) in the domain:
- At x = 2, f''(2) = 0 (inflection point).
- f''(x) is less than 0 for x < 2="" and="" greater="" than="" 0="" for="" x="" /> 2, indicating that f''(x) is not positive throughout the domain S.
- Hence, Statement III is incorrect.
Statement IV: Local Minima
- Since there is a local maxima and f(x) is continuous and differentiable within the boundary points, there must be at least one local minima as well.
- Therefore, Statement IV is correct.
Conclusion
Based on the evaluations:
- Statements I, II, and IV are correct, while III is incorrect.
- Thus, the correct answer is option D: Only statements I, II, and IV are correct.
Free Test
Community Answer
Consider the polynomial f[x) = x3 - 6x2 + 11x - 6 on the domain S, giv...
Given: f{x) = x3 - 6x2 + 11x  - 6
= (x - 1) (x - 2) (x - 3)     ...... (i)
  • At x = 1 and x = 3 f(x) = 0
    ∴ Statement 1 is correct
    f' (x) = 3x3 - 12x2 + 11
    f '(x) = 6x - 12 = 6 (x - 2)
  • Now, f'(x) < 0, if x < 2. This is condition for maxima which is, within the domin S.
  • Now, f '(x) > 0, if x > 2. This is condition for minima which is, within the domin S.
  • But, f'(x) is not greater than zero throughout the domains S.
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Consider the polynomial f[x) = x3 - 6x2 + 11x - 6 on the domain S, given by 1 ≤ x ≤ 3. The first and second derivatives are f(x) and f{x).Consider the following statements:I. The given polynomial is zero at the boundary points x = 1 and x = 3.II. There exists one local maxima of f{x) within the domain S.III. The second derivative f"(x) > 0 throughout the domains S.IV. There exists one local minima f(x) within the domain S.a)Only statements II and IV are correct.b)Only statements I and IV are correct.c)Only statements I. II and III are correct.d)Only statements I. II and IV are correct.Correct answer is option 'D'. Can you explain this answer?
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Consider the polynomial f[x) = x3 - 6x2 + 11x - 6 on the domain S, given by 1 ≤ x ≤ 3. The first and second derivatives are f(x) and f{x).Consider the following statements:I. The given polynomial is zero at the boundary points x = 1 and x = 3.II. There exists one local maxima of f{x) within the domain S.III. The second derivative f"(x) > 0 throughout the domains S.IV. There exists one local minima f(x) within the domain S.a)Only statements II and IV are correct.b)Only statements I and IV are correct.c)Only statements I. II and III are correct.d)Only statements I. II and IV are correct.Correct answer is option 'D'. Can you explain this answer? for Civil Engineering (CE) 2024 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about Consider the polynomial f[x) = x3 - 6x2 + 11x - 6 on the domain S, given by 1 ≤ x ≤ 3. The first and second derivatives are f(x) and f{x).Consider the following statements:I. The given polynomial is zero at the boundary points x = 1 and x = 3.II. There exists one local maxima of f{x) within the domain S.III. The second derivative f"(x) > 0 throughout the domains S.IV. There exists one local minima f(x) within the domain S.a)Only statements II and IV are correct.b)Only statements I and IV are correct.c)Only statements I. II and III are correct.d)Only statements I. II and IV are correct.Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the polynomial f[x) = x3 - 6x2 + 11x - 6 on the domain S, given by 1 ≤ x ≤ 3. The first and second derivatives are f(x) and f{x).Consider the following statements:I. The given polynomial is zero at the boundary points x = 1 and x = 3.II. There exists one local maxima of f{x) within the domain S.III. The second derivative f"(x) > 0 throughout the domains S.IV. There exists one local minima f(x) within the domain S.a)Only statements II and IV are correct.b)Only statements I and IV are correct.c)Only statements I. II and III are correct.d)Only statements I. II and IV are correct.Correct answer is option 'D'. Can you explain this answer?.
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