Consider the function $$f(x) = \sum_{k=1}^{m} (x-k)^4, \quad x \in \mathbb{R}$$ where $m > 1$ is an integer. Show that $f$ has a unique minimum and find the point where the minimum is attained.
Consider the function
$$f(x) = \sum_{k=1}^{m} (x-k)^4, \quad x \in \mathbb{R}$$
where $m > 1$ is an integer. Show that $f$ has a unique minimum and find the point where the minimum is attained.