Given $x \in [0,1]$, let $f_k(x)$ be concave function in $x$ for all $k= 0,1,...,N$. Is the following two maximization formulations equivalent? $$ \max_{0 \le k \le N} f_k(x) =?= \max \{f_0(x), f_1(x),...,f_N(x)\} $$
Here is my thinking, the Left Hand Side (LHS) is to find $k=k^*$ so that maximizes $f_k(x)$. Hence, the LHS eventually gives only one function $f_{k^*}(x)$ which is concave again. But the Right Hand Side is kind of using "maximum-intersection" over the functions $\{f_0(x),...,f_N(x)\}$, so the concavity may not be preserved. Hence the LHS is not equivalent to RHS. Is this correct way to go?
Any suggestion is appreciated. Thank you.
No, your thinking is not correct; the LHS and RHS describe exactly the same function of $x$ in different ways.
The problem with your interpretation of the LHS is that $x$ is not yet known. It is impossible for the LHS to return just one function $f_{k^*}(x)$, because the proper choice of $k^*$ depends on $x$.
The two expressions are exactly equivalent. At any fixed point $x$, the value of both expressions is the maximum of the $N$ values $f_0(x)$, $f_1(x)$, ..., $f_N(x)$.