Question

Calc max {a, (a+b)/2}+min{(a+b)/2, b} pt a,b, apartinand lui [0,infinit)

Answer 1

   
[tex]\displaystyle\\ \text{Calculati:}\\\\ \max \left\{a,~\frac{a+b}{2}\right\}+\min\left\{\frac{a+b}{2},~b\right\} ~pt~ a,b\in [0,~\infty)\\\\\\ \frac{a+b}{2} ~~\text{este media aritmetica a numerelor a si b.}\\\\ \text{Media aritmetica a 2 numere este situata intre cele 2 numere }\\ \text{daca cele 2 numere sunt diferite, iar daca numerele sunt egale, }\\ \text{atunci media aritmetica este egala cu fiecare din cele 2 numere.} \\\\ \text{Cazul 1:}\\ a = b ~~\Longrightarrow~~ \frac{a+b}{2} = a = b [/tex]


[tex]\displaystyle\\ \max \left\{a,~\frac{a+b}{2}\right\}+\min\left\{\frac{a+b}{2},~b\right\} = \boxed{a + b}\\\\ \text{Cazul 2:}\\ a \ \textless \ b ~~\Longrightarrow~~a\ \textless \ \frac{a+b}{2} \ \textless \ b\\\\ \max \left\{a,~\frac{a+b}{2}\right\}+\min\left\{\frac{a+b}{2},~b\right\} = \frac{a+b}{2}+\frac{a+b}{2}=\\\\=\frac{a+b+a+b}{2} =\frac{2a+2b}{2}=\frac{2(a+b)}{2} = \boxed{a + b}\\\\ \text{Cazul 3:}\\a\ \textgreater \ b~~\Longrightarrow~~a\ \textgreater \ \frac{a+b}{2} \ \textgreater \ b\\\\ \max\left\{a,~\frac{a+b}{2}\right\}+\min\left\{\frac{a+b}{2},~b\right\} =\boxed{a+b}[/tex]