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Part BCSIR NET December 2023rolle-counting-roots

Rolle counting roots

Let f(x) be a cubic polynomial with real coefficients. Suppose that f(x) has exactly one real root and that this root is simple. Which one of the following statements holds for ALL antiderivatives F(x) of f(x)?

  1. A.F(x) has exactly one real root.
  2. B.F(x) has exactly four real roots.
  3. C.F(x) has at most two real roots.✓
  4. D.F(x) has at most one real root.

Solution

F′ = f changes sign exactly once, so F is monotone on each side of one point: at most two real roots (and the constant of integration can realise 0, 1 or 2).

The trap it tests

Execution slip

The idea was right. The computation was not.

Drill statements like this

Related counterexample: Differentiable ⇒ continuously differentiable

More on this topic

From Continuity and Differentiation › Differentiability, mean value theorems, Taylor, L'Hôpital

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