Biggest number of components in a graph?
Suppose that a graph having $n$ vertices and $m$ edges consists of $k$ connected components, say of sizes $n_1,\ldots,n_k$. Then
$$
m \leq \binom{n_1}{2} + \cdots + \binom{n_k}{2}.
$$
Conversely, given $n_1,\ldots,n_k$ summing to $n$ which satisfy this inequality, we can construct a graph with $n$ vertices, $m$ edges, and at least $k$ connected components. We conclude that the maximum number of connected components of a graph with $n$ vertices and $m$ edges is the maximal $k$ such that we can find $n_1,\ldots,n_k \geq 1$ summing to $n$ satisfying the inequality above.

This leads to the following question: given $k$, maximize $\binom{n_1}{2} + \cdots + \binom{n_k}{2}$ subject to the constraint that $n_1,\ldots,n_k \geq 1$ are integers summing to $n$. The key is the following observation:
$$
\binom{a+1}{2} + \binom{b-1}{2} - \binom{a}{2} - \binom{b}{2}  = a-b+1.
$$
This shows that as long as $n_i \geq n_j$, moving a vertex from $n_j$ to $n_i$ increases $\binom{n_i}{2} + \binom{n_j}{2}$. It follows that the choice of $n_1,\ldots,n_k$ maximizing $\binom{n_1}{2} + \cdots + \binom{n_k}{2}$ is $n_1=n-k+1$, $n_2=\cdots=n_k=1$.

We have proved: the maximal number of connected components of a graph with $n$ vertices and $m$ edges is the maximal $k$ such that $$ m \leq \binom{n-k+1}{2}. $$