[14 points] For a positive integer $n$, let $f(x) = \sum_{i=0}^{n} x^i = 1 + x + x^2 + \cdots + x^n$. Find the number of local maxima of $f(x)$. Find the number of local minima of $f(x)$. For each maximum/minimum $(c, f(c))$, find the integer $k$ such that $k \leq c < k+1$.
Hints (use these or your own method): It may be helpful to (i) look at small $n$, (ii) use the definition of $f$ as well as a closed formula, and (iii) treat $x \geq 0$ and $x < 0$ separately.