Prove that Max (a,b) is equal consider three separate cases: a=b;a>b; and b>a

Prove that:
Max(a,b) = a+b l a-b l / 2

I will assume that you mean [(a+b) + la-bl]/2 and not (a+b) + (la-bl/2); otherwise, the equation would not always be true.

Note that Max(a, b) equals a if a > b, a if a = b, and b if b > a.
Note that |a - b| equals a - b if a > b, 0 if a = b, and b - a if b > a.

In each case, I will start off with the right-hand side and end with the left-hand side of the equation we need to prove.

If a > b, then [(a+b) + la-bl]/2 = [(a+b) + (a-b)]/2 = 2a/2 = a = Max(a, b).
If a = b, then [(a+b) + la-bl]/2 = [(a+a) + 0]/2 = 2a/2 = a = Max(a, b).
If b > a, then [(a+b) + la-bl]/2 =[(a+b) + (b-a)]/2 = 2b/2 = b = Max(a, b).

So in all cases, Max(a, b) = [(a+b) + la-bl]/2 .

Lord bless you today!