This is Exercise 8.5 in Lee's Riemannian Manifolds: An Introduction to Curvature.
Suppose $M\subset \mathbb R^{n+1}$ is a hypersurface with the induced metric. Let $p\in M$, and let $\kappa_1,\dots,\kappa_n$ denote the principal curvatures of $M$ at $p$ with respect to some choice of unit normal.
Show that $M$ can be approximated locally by the quadratic polynomial $\frac{1}{2} h$ [the scalar second fundamental form] in the following sense: There are Euclidean coordinates $(x,y)=(x^1,\dots, x^n, y)$ centered at $p$ such that $M$ is described locally by an equation of the form $y=f(x)$, where the second-order Taylor series of $f$ at the origin is $$f(x)=\frac{1}{2}(\kappa_1(x^1)^2)+\cdots+\kappa_n(x^n)^2)+O(|x|^3).$$
If $n=2$, show that $\kappa_1$ and $\kappa_2$ are equal to the minimum and maximum signed Euclidean curvatures of $M$-geodesics passing through $p$, and also to the minimum and maximum signed Euclidean curvatures of plane curves obtained by intersection $M$ with planes orthogonal to $T_p M$.
I'm stuck on the first part. Let's consider the problem with $n=2$, as the ideas here almost certainly generalize. My idea is to place $p$ at the origin with the normal vector pointing along the $z$-axis. Then we should have a map: $(x,y)\rightarrow (x,y,f(x,y))$ given by projection along the $z$-axis. But I'm having trouble making this precise. In particular, I don't see why this must be well-defined (the projection could never intersect the submanifold) or smooth. I suspect this ultimately ends up being some inverse function theorem trick.
But this not the whole story. Recall the $\kappa_i$ are defined as eigenvalues of the operator $s$ on $T_pM$, so we probably need to do a change of coordinates on $(x,y)$ to ensure the $\kappa_i$ are sent to the eigenvectors $E_i$.
If that map is given, clearly the way to proceed is to do a Taylor expansion and show we get the desired coefficients. It seems that the way we chose the coordinate system should force all the first derivatives to be zero. I am lost on how to get the second order terms.
Any help is greatly appreciated. I have sketched an attempt in the comments that I think works.