Given $F= \{ f(x) \text{ is continuous on } [0, 1]$ and $f$ has a local extrema at $x=1/2 \}$, How to prove or disprove that $F$ is a vector space

Question

I tried it by considering the cases individually. First, take $f(1/2)$ as the maximum, and then take $f(1/2)$ as the minimum ,I can prove that they are not vector spaces, but I am confused about the term 'extrema.' Disproving one case of maxima or minima as not being a vector space may validate the other one or vice versa.Please help me in this regard

Answer 1

Pick a function $f$ without a local extremum at $\frac12$, then subtract from it a function $g$ with a local extremum at $\frac12$. If you choose the functions carefully, $f - g$ can have a local extremum at $\frac12$, but $(f - g) + g = f$ does not. As a hint, if the functions were not required to be continuous you could choose $f(x) = x$ and $g(x)$ equal to $1$ at $x = \frac12$ and $0$ otherwise.