Let $(M,g)$ be a compact Riemannian manifold and let ${\rm sec}(X,Y)$ denote sectional curvature of plane spanned by $X,Y\in T_pM$. Then
How to prove or disprove that $f(p)$ defined by $$f(p):=\min\{{\rm sec}(X,Y)\,\,|\,\, X,X\neq Y\in T_pM\},$$ is a smooth function on $M$?
It's not, in general. For example, you could let $M = \mathbb R^2 \times N$ with a product metric, where $\mathbb R^2$ has the Euclidean metric and $N$ is a surface of revolution whose Gaussian curvature transitions smoothly from positive to negative, such as the surface illustrated below. With a little care, you could make the Gaussian curvature of the surface equal to something like $-z$, and then the minimum sectional curvature of the product manifold would be $\min(-z,0)$, which is not smooth.