  Let $f$ be a real polynomial function with $n$ variables and $S$ be a basic
closed semialgebraic set in $\Bbb{R}^n$. In this paper, we are interested in
the problem of identifying the type (local minimizer, maximizer or not extremum
point) of a given isolated KKT point $x^*$ of $f$ over $S.$ To this end, we
investigate some properties of the tangency variety of $f$ on $S$ at $x^*,$ by
which we introduce the definition of faithful radius of $f$ over $S$ at $x^*.$
Then, we show that the type of $x^*$ can be determined by the global extrema of
$f$ over the intersection of $S$ and the Euclidean ball centered at $x^*$ with
a faithful radius. Finally, we propose an algorithm involving algebraic
computations to compute a faithful radius of $x^*$ and determine its type.
