If $f, g : [a,b] \to\mathbb{R}$ are integrable then the following functions are also integrable $$h : [a, b] \to\mathbb{R} \text{ with } h(x) = \max\{f(x), g(x)\}.$$ $$T: [a, b] \to\mathbb{R} \text{ with } T(x) = \min\{f(x), g(x)\}.$$
For h:
It is worth that $$\max(a,b)=\frac{a+b+|a-b|}{2}$$
so
$$\max\{(x), g(x)\} = \frac{f(x) + g(x) + \mid f(x) - g(x)\mid}{2}$$
hence $\max\{f(x), g(x)\}$ is integrable because the absolute value is integrable.
For T:
It is worth that $$\min(a,b)=\frac{a+b-|a-b|}{2}$$
so
$$\min\{(x), g(x)\} = \frac{f(x) + g(x) - \mid f(x) - g(x)\mid}{2}$$
hence $\min\{f(x), g(x)\}$ is integrable because the absolute value is integrable.
Is my attempt correct/complete? It looks like a bit simple, but I can't think anything else to add. I appreciate in advance any assistance.