Vector Space or Not?

Question

I have a question. Suppose that $V$ is a set of all real valued functions that attain its relative maximum or relative minimum at $x=0$. Is V a vector space under the usual operations of addition and scalar multiplications? My guess is it is not a vector space, but I can't able to give a counterexample?

Answer 1

As I hinted in my comment above, the terms local maximum and local minimum only really make sense when talking about differentiable functions. So here I show that the set of functions with a critical point (not necessarily a local max/min) at 0 (really at any arbitrary point $a \in \mathbb{R}$) is a vector subspace.

If you broaden this slightly to say that $V \subset C^1(-\infty,\infty)$ is the set of differentiable functions on the reals that have a critical point at 0 (i.e. $\forall$ $f \in V$ $f'(0) = 0$).

Then it's simple to show that this is a vector space.

If $f,g \in V$ ($f'(0)=g'(0)=0$), and $r \in \mathbb{R}$ then, and hence

1. $(f+g)'(0) = f'(0) + g'(0) = 0 + 0 = 0$.
2. The derivative of the zero function is zero, and hence evaluates to 0 at 0.
3. $(rf)'(0) = r(f'(0))=r0 = 0$.

And together these imply that $V$ is a vector subspace of $C^1(-\infty,\infty)$.

Robert Israel's answer above is a nice example of why we must define our vector space to have a critical point, not just a max/min at 0.

Answer 2

Consider the functions $$f(x)=\cases{x&\text{if }x<0\\0&\text{otherwise}}$$ and $$g(x)=\cases{0&\text{if }x<0\\x&\text{otherwise.}}$$

Answer 3

For example, $x^2 + x^3$ and $x^2$ both have relative minima at $0$, but $(x^2 + x^3) - x^2$ does not.