Explicație pas cu pas:
2.a)
[tex]E(x) = \Big( \frac{1}{x + 2} - \frac{1}{2x + 1}\Big) : \frac{1}{2 {x}^{2} + 5x + 2} = \\ = \frac{2x + 1 - (x + 2)}{(x + 2)(2x + 1)} \cdot (2 {x}^{2} + 5x + 2) \\ = \frac{2x + 1 - x - 2}{2 {x}^{2} + 5x + 2} \cdot (2 {x}^{2} + 5x + 2) = \bf x - 1[/tex]
b)
[tex]E(-x) = - x - 1 = - (x + 1)[/tex]
[tex]E(-x) \cdot E(x) = - (x + 1)(x - 1) = - {x}^{2} + 1 \\ [/tex]
[tex]{x}^{2} \geqslant 0 \iff \ - {x}^{2} \leqslant 0 \: | + 1 \\ \implies - {x}^{2} + 1 \leqslant 1 [/tex]
[tex]\implies \bf E(-x) \cdot E(x) \leqslant 1[/tex]
3.a)
[tex]f(a) = b \iff a - 5 = b \\ 3a = 2b[/tex]
[tex]3a = 2(a - 5) \iff 3a = 2a - 10 \\ \implies \bf a = - 10 \\ b = ( - 10) - 5 = - 10 - 5 \implies \bf b = - 15[/tex]
b) notăm OM mediana
[tex]OA = 5 \: u.m., OB = 5 \: u.m. \implies AB = 5 \sqrt{2} \: u.m. \\ [/tex]
[tex]OM \cdot AB = OA \cdot OB \\ OM \cdot 5 \sqrt{2} = 5 \cdot 5 \implies \bf OM = \frac{5 \sqrt{2} }{2} \: u.m.[/tex]
