  Let $C_b(K)$ be the set of all bounded continuous (real or complex) functions
on a complete metric space $K$ and $A$ a closed subspace of $C_b(K)$. Using the
variational method, it is shown that the set of all strong peak functions in
$A$ is dense if and only if the set of all strong peak points is a norming
subset of $A$. As a corollary we show that if $X$ is a locally uniformly
convex, complex Banach space, then the set of all strong peak functions in
$\mathcal{A}(B_X)$ is a dense $G_\delta$ subset. Moreover if $X$ is separable,
smooth and locally uniformly convex, then the set of all norm and numerical
strong peak functions in $\mathcal{A}_u(B_X:X)$ is a dense $G_\delta$ subset.
In case that a set of uniformly strongly exposed points of a (real or complex)
Banach space $X$ is a norming subset of $\mathcal{P}({}^n X)$ for some $n\ge
1$, then the set of all strongly norm attaining elements in $\mathcal{P}({}^n
X)$ is dense, in particular, the set of all points at which the norm of
$\mathcal{P}({}^n X)$ is Fr\'echet differentiable is a dense $G_\delta$ subset.
