Question

La 7) Raspunsul corect poate sa fie D)? Nu sunt sigur pe raspuns si as avea nevoie de ajutor..!

Answer 1

$f(x) = x^3+ax \\ \\\\ f'(x) = 0 \Rightarrow 3x^2+a = 0 \Rightarrow 3x^2 = -a \Rightarrow x^2 = -\dfrac{a}{2}\Rightarrow \\ \Rightarrow x = \pm \sqrt{\dfrac{-a}{2}} \\ \\\\ \text{Deoarece functia vine de la }-\infty \text{ la }+\infty,\\\\x = -\sqrt{-\dfrac{a}{2}}\text{ este punct de maxim local, iar}\\ x = \sqrt{-\dfrac{a}{2}}\text{ este punct de minim local.}$

$\text{Conditia este: }f\Bigg(\pm\sqrt{\dfrac{-a}{2}}\Bigg) = 0 \Rightarrow \\ \\ \Rightarrow \pm\Big(\dfrac{-a}{2}\Big)\sqrt{\dfrac{-a}{2}}\pm a\sqrt{\dfrac{-a}{2}} = 0 \Rightarrow \mp\dfrac{a}{2}\pm a = 0 \Rightarrow \\ \\ \Rightarrow \mp a\pm 2a = 0 \Rightarrow \boxed{a = 0}$