Question

Demonstreaza ca -2≤E(x)≤2, pentru orice x€R-{-1,1}. E(x)= 4x/x²+1​

Answer 1

Explicație pas cu pas:

$E(x) - 2= \frac{4x}{ {x}^{2} + 1 } - 2 = \frac{4x - 2 {x}^{2} - 2 }{ {x}^{2} + 1 } = \frac{ - 2({x}^{2} - 2x + 1) }{ {x}^{2} + 1 } = \frac{ - 2(x - 1)^{2} }{ {x}^{2} + 1 } \leqslant 0$

$E(x) - 2 \leqslant 0 = > E(x) \leqslant 2$

$E(x) + 2 = \frac{4x}{ {x}^{2} + 1 } + 2 = \frac{4x + 2 {x}^{2} + 2}{ {x}^{2} + 1 } = \frac{ 2({x}^{2} + 2x + 1) }{ {x}^{2} + 1 } = \frac{ 2(x + 1)^{2} }{ {x}^{2} + 1 } \geqslant 0$

$E(x) + 2 \geqslant 0 = > E(x) \geqslant - 2$

deci:

$- 2 \leqslant E(x) \leqslant 2$

pentru orice x€R-{-1,1}.