Multiple choice

Equation $\displaystyle (x-a)^{3}+(x-b)^{3}+(x-c)^{3}+(x-d)^{3}+(x-d)^{3}=0$ has (where $a, b, c, d$ are not all equal)

  1. Exactly one real root

  2. At least one real root

  3. All real root

  4. No real root
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A Correct answer
Explanation

The sum of the five cubic terms is a strictly increasing function of x, because its derivative is a sum of squared terms and is positive unless all constants are equal. Since the constants are not all equal, the equation crosses zero exactly once. Therefore, it has exactly one real root.

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