Control the sectional curvature

I think you might be asking the following: Suppose $M$ is a hypersurface in Euclidean space with the Riemannian metric induced by the Euclidean structure. Suppose that at a point $p$, there exists a sphere $S$ that is tangent to $M$ at $p$, and is such that there exists a neighborhood $V \subset M$ of $p$ that lies inside $S$. Then are all the sectional curvatures of $M$ at $p$ greater than the sectional curvature of $S$?

The answer to this question is yes.

One way to prove this is to show that the second fundamental form $H_M$ of $M$ at $p$ satisfies $$ H_M \ge \frac{1}{R}I, $$ where $R$ is the radius of the sphere.