Question

Daca numerele x,y satisfac relatia |x+y|+|x-y|=2, care este valoarea maxima a expresiei x^2 -6x+ y^2 ?

Answer 1

$|x+y|+|x-y|=2 \\\\ \sqrt{(x+y)^2}+\sqrt{(x-y)^2} = 2\Big|^2\\ \\ (x+y)^2+(x-y)^2+2\sqrt{[(x+y)(x-y)]^2} = 4 \\ (x+y)^2+(x-y)^2+2\sqrt{(x^2-y^2)^2}=4 \\x^2+2xy+y^2+x^2-2xy+y^2+2|x^2-y^2| = 4 \\ 2(x^2+y^2)+2|x^2-y^2| = 4 \\ x^2+y^2+|x^2-y^2| = 2 \\ x^2+y^2 = 2-|x^2-y^2| \\ \\ E(x,y) = x^2-6x+y^2$

$x^2-6x+y^2 = 2-|x^2-y^2|-6x\\ \\ \text{Pentru ca }x^2-6x+y^2 \text{ sa fie maxim trebuie obligatoriu ca: }$

$|x^2-y^2| = 0 \Rightarrow x^2 = y^2 \Rightarrow x = \pm y \\ \\ |\pm y+y|+|\pm y-y| = 2 \Rightarrow \\ (1)\quad 2|y| = 2 \Rightarrow y = \pm 1\\ (2)\quad |-2y|=2 \Rightarrow y = \pm 1\\ \\\Rightarrow x = 1\quad sau\quad x = -1 \\ \\ \Rightarrow E(1,- 1) = 2-6 = -4\\ \\\Rightarrow E(-1,1) = 2+6 = 8 \\ \\ \Rightarrow E(x,y)_{max} = 8$