[tex]f:\mathbb{R}\to \mathbb{R},\quad f(x) = \dfrac{a^2}{2}x+a,\quad a\neq 0\\ \\ A:\quad f(0) = a \Rightarrow A(0,a) \\ \\B:\quad f(x) = 0 \Rightarrow \dfrac{a^2}{2}x+a = 0\Rightarrow ax+2 = 0 \Rightarrow B\Big(-\dfrac{2}{a},0\Big)\\ \\ \\ AB =\sqrt{\Big(-\dfrac{2}{a}-0\Big)^2+(0-a)^2} = \sqrt{\dfrac{4}{a^2}+a^2} =\\ \\ =\sqrt{\underbrace{\Big(\dfrac{2}{a}-a\Big)^2}\limits_{\geq 0}+4}\geq \sqrt{0+4}=\sqrt{4} = 2\\ \\ \\ \Rightarrow AB \geq 2,\quad \forall a\in \mathbb{R}^*[/tex]
