Question

Let f : \(\R → \R\) be an infinitely differentiable function such that f" has exactly two distinct zeroes. Then

• A. f' has at most 3 distinct zeroes
• B. f' has at least 1 zero
• C. f has at most 3 distinct zeroes
• D. f has at least 2 distinct zeroes

Answer 1

The Correct Option is A

To solve this problem, we need to analyze the behavior of the function \( f \) and its derivatives given the conditions of the problem. The function \( f : \R \to \R \) is infinitely differentiable, and we know that its second derivative \( f'' \) has exactly two distinct zeroes.

The properties of the function and its derivatives can be linked to the concepts in calculus related to critical points and inflection points. Here's a step-by-step explanation:

1. Understanding \( f'' \):
   • The function \( f''(x) \) having exactly two zeroes means that the concavity of \( f(x) \) changes at exactly two points. These points are inflection points, where the curvature changes direction.
2. Impact on \( f' \):
   • The zeroes of \( f''(x) \) imply critical points of \( f'(x) \). Between any two zeroes of \( f'' \), \( f' \) can have local extrema (maxima or minima).
   • By Rolle's Theorem or Mean Value Theorem, a polynomial segment defined between two zeroes of \( f'' \) can have at most one zero in this interval.
3. Conclusion for \( f' \):
   • If \( f''(x) \) has exactly two zeroes, say at \( x_1 \) and \( x_2 \), then between \( x_1 \) and \( x_2 \), \( f' \) can have at most one zero, and hence all these together allow \( f' \) to have at most three distinct zeroes (consider a zero potentially outside these intervals as well).
4. Ruling out other options:
   • Since \( f' \) is at most three distinct zeroes, it does not assure \( f(x) \) will have any zero; so "f has at least 2 distinct zeroes" is unsubstantiated.
   • "f has at most 3 distinct zeroes" directly doesn't follow since it's about the original function \( f(x) \), not its derivative traits linked to the curvature.
   • "f' has at least 1 zero" cannot be confidently generalized without specific initial constraints, in light of given conditions.

Therefore, the correct choice that satisfies these conditions is that f' has at most 3 distinct zeroes.