if two functions attain maximum at common point, will the sum of the two functions also attain maximum at the common point?

Question

Let $X$ be a compact hausdorff space and $E$ be a Banach space. Suppose that $f,g:X \rightarrow E$ are continuous Banach-space valued functions. Denote the sup-norm on $E$ as $\| f \| = \sup_{x \in X}\| f(x) \|.$

Question: If $\|f\|=\|f(x_0)\|$ and $\|g\| = \|g(x_0)\|$ for some $x_0 \in X,$ is it true that $\|(f+g)\| = \|(f+g)(x_0)\|?$ In other words, if two functions attain maximum at common point, will the sum of the two functions also attain maximum at the common point?

This seems quite intuitive from a graph. But I do now know a proof of it.

Any hint would be appreciated.