  Let $\mathcal{H}_{n,d} := \mathbb{R}[x_1$,$\ldots$, $x_n]_d$ be the set of
all the homogeneous polynomials of degree $d$, and let $\mathcal{H}_{n,d}^s :=
\mathcal{H}_{n,d}^{\mathfrak{S}_n}$ be the subset of all the symmetric
polynomials. For a semialgebraic subset of $A \subset \mathbb{R}^n$ and a
vector subspace $\mathcal{H} \subset \mathcal{H}_{n,d}$, we define a PSD cone
$\mathcal{P}(A$, $\mathcal{H})$ by $\mathcal{P}(A$, $\mathcal{H}) := \big\{f
\in \mathcal{H}$ $\big|$
  $f(a) \geq 0$ ($\forall a \in A$)$\big\}$. In this article, we study a family
of extremal symmetric polynomials of $\mathcal{P}_{3,6} :=
\mathcal{P}(\mathbb{R}^3$, $\mathcal{H}_{3,6})$ and that of $\mathcal{P}_{4,4}
:= \mathcal{P}(\mathbb{R}^4$, $\mathcal{H}_{4,4})$. We also determine all the
extremal polynomials of $\mathcal{P}_{3,5}^{s+} := \mathcal{P}(\mathbb{R}_+^3$,
$\mathcal{H}_{3,5}^s)$ where $\mathbb{R}_+ := \big\{ x \in \mathbb{R}$, $x \geq
0 \big\}$. Some of them provide extremal polynomials of $\mathcal{P}_{3,10}$.
