Let $(f_n)$ be a sequence of twice differentiable functions on $\mathbb R$ such that for each $n$ there exists some $x_n\in\mathbb{R}$ such that:

• $f_n$ is strictly convex on $(-\infty,x_n)$,
• $f_n$ is strictly concave on $(x_n, +\infty)$.

Suppose also that $f_n$ uniformly converges to a twice differentiable function $f$.

It was conjectured that then the sequence $(x_n)$ will be convergent.

This conjecture was disproved.

The OP then asked in a comment whether the additional condition that $f''$ not vanish on any nonempty interval can help.

Below it will be shown that the answer is still negative, even with the latter additional condition.

Let $(f_n)$ be a sequence of twice differentiable functions on $\mathbb R$ such that for each $n$ there exists some $x_n\in\mathbb{R}$ such that:

• $f_n$ is strictly convex on $(-\infty,x_n)$,
• $f_n$ is strictly concave on $(x_n, +\infty)$.

Suppose also that $f_n$ uniformly converges to a twice differentiable function $f$.

It was conjectured that then the sequence $(x_n)$ will be convergent.

This conjecture was disproved.

The OP then asked in a comment whether the additional condition that $f''$ not vanish on any nonempty interval can help.

Below it will be shown that the answer is still negative, even with the latter additional condition.

In another answer, it will be shown how conditions on the $f_n$'s or on $f$ can be modified to make the sequence $(x_n)$ convergent.

Let $(f_n)$ be sequence of twice differentiable functions on $\mathbb R$ such that for each $n$ there exists some $x_n\in\mathbb{R}$ such that:

• $f_n$ is strictly convex on $(-\infty,x_n)$,
• $f_n$ is strictly concave on $(x_n, +\infty)$.

Suppose also that $f_n$ uniformly converges to a twice differentiable function $f$.

It was conjectured that then the sequence $(x_n)$ will be convergent.

This conjecture was disproved.

The OP then asked in a comment whether the additional condition that $f''$ not vanish on any nonempty interval can help.

Below it will be shown that the answer is still negative, even with the latter additional condition.

Let $(f_n)$ be a sequence of twice differentiable functions on $\mathbb R$ such that for each $n$ there exists some $x_n\in\mathbb{R}$ such that:

• $f_n$ is strictly convex on $(-\infty,x_n)$,
• $f_n$ is strictly concave on $(x_n, +\infty)$.

Suppose also that $f_n$ uniformly converges to a twice differentiable function $f$.

It was conjectured that then the sequence $(x_n)$ will be convergent.

This conjecture was disproved.

The OP then asked in a comment whether the additional condition that $f''$ not vanish on any nonempty interval can help.

Below it will be shown that the answer is still negative, even with the latter additional condition.