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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 domain S. IV. There exists one local minima of f(x) within the domain S. The correct option is:

1. Only statements I, II and III are correct.
2. Only statements I, II and IV are correct.
3. Only statements I and IV are correct.
4. Only statements II and IV are correct.

Correct answer: Only statements I, II and IV are correct.

Solution

Statements I, II, and IV are correct because the polynomial evaluates to zero at the endpoints, has one local maximum, and one local minimum within the given domain. Statement III is incorrect as the second derivative does not remain positive throughout the entire interval.

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