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Ce qui donne : 1. ∀x ∈ ℝ f(x) = x(x − 2\sqrt{3})(x + 2\sqrt{3}).

2. Un produit de facteur est nul si, et seulement si, au moins est de ses facteurs est nul :

f(x) = 0 ⇔ x ∈ \{−2\sqrt{3}; 0; 2\sqrt{3}\}

3. Tableau de signes ci-contre :

f(x) < 0 ⇔ x ∈] − ∞; −2\sqrt{3}[∪]0; 2\sqrt{3}[

x x x^{2} − 12 f(x) −∞ −2\sqrt{3} 0 0  0 0 0  2\sqrt{3} 0 0  +∞ −  −  +  + +  −  −  + −  +  −  +

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x f(x) 0_{\frac{3}{2}}  1 +∞  ^{\frac{3}{2}}0  +∞ 1

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t {Sgn. \atop x′(t)} \begin{matrix}Var. \\ x \end{matrix} \begin{matrix}Var. \\ y \end{matrix} {Sgn. \atop y′(t)} T⃗ m_{T}  0 1, 5 2 0 1, 5^{(\begin{matrix}1, 5 \\ 1, 5 \end{matrix})} 1  1 0 4 4 3^{(\begin{matrix}0 \\ 3 \end{matrix})} ∞  2 −1.5 2 6 0^{(\begin{matrix}−1.5 \\ 0 \end{matrix})} 0  3 0 0 4 −2^{(\begin{matrix}0 \\ −2 \end{matrix})} ∞  4 1 2 0 1, 5^{(\begin{matrix}1 \\ −1 \end{matrix})} −1 + +  − +  −  + −

4