Analyzing Real Roots of a Cubic Function

Very Hard Problem Solving Algebra

Let $a$ and $b$ be two distinct real numbers. Consider the polynomial function:

$$ f(x) = x^3 - 3ax^2 + 3bx - ab $$

Which of the following statements is true regarding the number of real roots of $f(x)$?

The function has exactly one real root.

The function has exactly two real roots.

The function has three real roots.

The function has either one or three real roots.

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