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The problem you've uploaded involves a continuous function \( f(x) \) and asks to show the existence of a point \( c \in (a, b) \) such that \( f''(c) = 0 \). It also asks to find an example of a function \( f(x) \) and an interval \( (a, b) \) such that this result holds.

Let's break down the problem:

1. **Given Information:**
   - The function \( f(x) \) is continuous on the closed interval \([a, b]\) and twice differentiable on the open interval \((a, b)\).
   - \( f(a) = f'(a) = f(b) \).

2. **Part 1: Show the existence of a point \( c \in (a, b) \) such that \( f''(c) = 0 \).**
   - This is a classic application of Rolle's Theorem, which states that if a function is continuous on a closed interval \([a, b]\), differentiable on the open interval \((a, b)\), and if \( f(a) = f(b) \), then there exists at least one point \( c \in (a, b) \) where \( f'(c) = 0 \).
   - Here, you're also given that \( f'(a) = f(b) \), so you apply Rolle's Theorem again to the first derivative \( f'(x) \), implying that there exists a point \( c \in (a, b) \) where \( f''(c) = 0 \).

3. **Part 2: Find an example of a function \( f(x) \) and a corresponding interval \([a, b]\) such that the result holds.**
   - An example of such a function could be \( f(x) = x^3 - 3x + 2 \). You would then find an interval \([a, b]\) for which the given conditions apply, and verify that \( f''(c) = 0 \) at some point in that interval.

Let me know if you would like to proceed with solving this or need more clarification.

La f(x) være en funksjon som er kontinuerlig for x ∈ [a, b], to ganger deriverbar for x ∈ (a, b), og der f(a) = f'(a) = f(b). Vis at det da finnes en c ∈ (a, b) slik at f''(c) = 0. Finn et eksempel på en funksjon f(x) og et tilhørende intervall (a, b) slik at resultatet over holder.

Explore the application of Rolle's Theorem to continuous and differentiable functions, including the existence of a point where the second derivative equals zero. This problem covers advanced calculus concepts and is suitable for students in Grades 11-12.

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