Explicație pas cu pas:
[tex]f(x) = - {x}^{2} + 4x + 3 = - {x}^{2} + 4x - 4 + 4 + 3 \\ = 7 - (x - 2)^{2} [/tex]
[tex]f(1 + \sqrt{2} ) = 7 - (1 + \sqrt{2} - 2)^{2} \\ = 7 - ( \sqrt{2} - 1)^{2}[/tex]
[tex]f(1 + \sqrt{3} ) = 7 - {(1 + \sqrt{3} - 2)}^{2} \\ = 7 - {(\sqrt{3} - 1)}^{2}[/tex]
[tex]f(3) = 7 - (3 - 2)^{2} = 7 - 1[/tex]
→
[tex]( \sqrt{2} - 1) < ( \sqrt{3} - 1) < 1 [/tex]
=>
[tex]f(3) < f(1 + \sqrt{3} ) < f(1 + \sqrt{2} )[/tex]
