Esercizio Risolvere la seguente disequazione intera:
$\dfrac{1}{2}x-\left[\left(\dfrac{x+1}{2}\right)^2-\left(\dfrac{x-1}{2}\right)^2\right]+\dfrac{1}{2}x^2\geq\left[\left(\dfrac{x+1}{2}\right)^2+\left(\dfrac{x-1}{2}\right)^2\right]^2-\dfrac{1}{4}x^4$
$\textbf {Soluzione}$
$\dfrac{1}{2}x-\left[\left(\dfrac{x+1}{2}-\dfrac{x-1}{2}\right)\left(\dfrac{x+1}{2}+\dfrac{x-1}{2}\right)\right]+\dfrac{1}{2}x^2\geq \left[\dfrac{x^2+2x+1}{4}+\dfrac{x^2-2x+1}{4}\right]^2-\dfrac{1}{4}x^4$
$\dfrac{1}{2}x-\left[\left(1\right)\left(x\right)\right]+\dfrac{1}{2}x^2\geq \left[\dfrac{x^2}{2}+\dfrac{1}{2}\right]^2-\dfrac{1}{4}x^4$
$\dfrac{1}{2}x-x+\dfrac{1}{2}x^2\geq \dfrac{1}{4}x^4 +\dfrac{1}{2}x^2+\dfrac{1}{4}-\dfrac{1}{4}x^4$, facendo le dovute semplificazioni si ottiene:
$\dfrac{1}{2}x-x\geq\dfrac{1}{4}\Longrightarrow 2x-4x\geq1\Longrightarrow -2x\geq1\Longrightarrow x\leq-\dfrac{1}{2}$ e dunque la soluzione è l'insieme $S=\left(-\infty,-\dfrac{1}{2}\right]$