We have a function $V(x)$ (potential energy) with $x$ being some variable. This function has a minimum at a certain $x_0$. We assume that $V(x)$ is an analytic real function of $x$ around $x_0$. Introduce the variable $\Delta x \equiv x - x_0$. For a sufficiently small $\Delta x$ this system will behave as a harmonic oscillator.
How can you show that this system will behave as a harmonic oscillator? It probably has to do with a Taylor series of $V(x)$, but that's all I know.
Perhaps this is better-suited to physics.se, but I'll anyway give an answer in the flavor of elementary differential calculus.
Hint Here we'll assume that $V$ is, say, $C^2$ in a neighborhood of $x_0$. Since $x$ has a minimum at $x_0$, the first and second derivative tests give that $V'(x_0) = 0$ and $a := V''(x_0) > 0$, so expanding the Taylor series for $V$ about $x = x_0$ (that is, $\Delta x = 0$) gives $$V(x) = a (\Delta x)^2 + O((\Delta x)^3) .$$ (For convenience we've normalized the potential energy so that $V(x_0) = 0$.)