Finding out the largest value of $m$ for the given $n$, such that $m^{3}\leq n$ is a paradigm of Binary search.
How it turns out to be Binary Search
1) Let’s fix what could be the range of values of $m$, i.e $1<=m<=n$
2) Say for some $K$, $K^{3}\leq n$ then the equation also holds for $K-1$(i.e $\left ( K-1 \right )^{3}\leq n$). Since we are looking for larger such $K$, then we start probing the values of $m>=K$.
3) Say, for some $K$, $K^{3}>n$ then the equation also doesn’t hold for $K+1$(i.e $\left ( K+1 \right )^{3}>n$). So we start probing the values of $m<K$.
These three points form the basis for any Binary Search Problem.
At any stage, we are reducing our search space by half.
Total time complexity is $O(logn)$