Question

Ordonaţi crescător numerele f(1+√2), f(1+√3) şi f(3), unde f:R→R, f(x)=-x² + 4x +3.​

Answer 1

Explicație pas cu pas:

$f(x) = - {x}^{2} + 4x + 3 = - {x}^{2} + 4x - 4 + 4 + 3 \\ = 7 - (x - 2)^{2}$

$f(1 + \sqrt{2} ) = 7 - (1 + \sqrt{2} - 2)^{2} \\ = 7 - ( \sqrt{2} - 1)^{2}$

$f(1 + \sqrt{3} ) = 7 - {(1 + \sqrt{3} - 2)}^{2} \\ = 7 - {(\sqrt{3} - 1)}^{2}$

$f(3) = 7 - (3 - 2)^{2} = 7 - 1$

→

$( \sqrt{2} - 1) < ( \sqrt{3} - 1) < 1$

=>

$f(3) < f(1 + \sqrt{3} ) < f(1 + \sqrt{2} )$