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.