Given the function $f:\mathbb{R}\rightarrow\mathbb{R}, f(x)=cos^{2}(\pi x)$ and the partition $P=\left\{ \frac{k}{n}\mid k=0,1,\ldots,2n\right\}$.
How can I calculate the upper sum U(f,P) of the function f(x) over the given partition P?
Similarly, how can I calculate the lower sum L(f,P) of the function f(x) over the partition P?
How to determine whether the sum, S, of the function f(x) over the partition P is an upper sum or a lower sum.
Any insights or step-by-step explanations on how to approach this problem would be greatly appreciated. Thank you!