argmax of sum of two functions given constraints on argmax of each of them

Suppose we have two single variable functions $$f(x)$$ and $$g(x)$$. Suppose further that

$$argmax\Big[f\Big]\in[a,b]\quad \&\quad argmax\Big[g\Big]\in[a,b]$$

where $$a,b\in\mathbb{R}$$, moreover, assume $$\max_x\Big[f(x)\Big]=\max_x\Big[g(x)\Big]$$

and also suppose that:

$$\exists\ C\ \subset[a,b]\quad s.t.\quad \forall x\in C\ ,\ \ f(x)=g(x)$$

And the above subset is the subset that contains all the points in which $$f(x)=g(x)$$.

1. can we conclude that we have the following? $$argmax\Big[f+g\Big]\in[a,b]$$

2. If the above conclusion is not generally correct, what can we say about $$argmax\Big[f+g\Big]$$?

Let $$f$$ be $$1+\frac{\epsilon}{2}$$ everywhere except $$f(a) = 1+7\epsilon$$ and $$f(C)=1$$
Let $$g$$ be $$1-\frac{\epsilon}{2}$$ everywhere except $$g(b) = 1+8\epsilon$$ and $$g(C)=1$$
Now add an extra peak somewhere outside the interval, for instance $$f(c) = 1+5\epsilon$$ and $$g(c) = 1+6\epsilon$$. $$f+g$$ is now maximized at $$c$$ which is arbitrary.