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Consider the polynomial $f(x) = x^{3} – 6x^{2} + 11x – 6$ on the domain $S$ given by $1 \leq x \leq 3.$ The first and second derivatives are $f’(x)$ and $f’’(x).$

Consider the following statements:

  1. The given polynomial is zero at the boundary points $x = 1$ and $x = 3.$
  2. There exists one local maxima of $f(x)$ within the domain $S.$
  3. The second derivative $f’’(x) > 0$ throughout the domain $S.$
  4. There exists one local minima of $f(x)$ within the domain $S.$

The correct option is:

  1. Only statements $\text{I, II}$ and $\text{III}$ are correct.
  2. Only statements $\text{I, II}$ and $\text{IV}$ are correct.
  3. Only statements $\text{I}$ and $\text{IV}$ are correct.
  4. Only statements $\text{II}$ and $\text{IV}$ are correct.
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