Wyznacz monotoniczność asymptoty i ekstrema funkcji

[m4rc3ll]

\(f(x) = x^2*e^{-x^2/2} \)

\(f'(x) = e^{-x^2/2}*(2x -x^3) \)

[eresh]

\(f(x) = x^2*e^{-x^2/2} \)

\(f'(x) = e^{-x^2/2}*(2x -x^3) \)

\(f'(x)>0\iff x(\sqrt{2}-x)(\sqrt{2}+x)>0\iff x\in(-\infty, -\sqrt{2})\cup (0,\sqrt{2})\\
f'(x)<0\iff x\in (-\sqrt{2},0)\cup (\sqrt{2},\infty)\\\)
funkcja rośnie w przedziałach \((-\infty, -\sqrt{2}), (0,\sqrt{2})\), maleje w \((-\sqrt{2},0), (\sqrt{2},\infty)\)
\(f_{max}=f(-\sqrt{2})=f(\sqrt{2})\\
f_{min}=f(0)\)
\(
\Lim_{x\to \infty}\frac{f(x)}{x}=\Lim_{x\to \infty}xe^{-\frac{x^2}{2}}=\Lim_{x\to \infty}\frac{x}{e^{\frac{x^2}{2}}}=\Lim_{x\to \infty}\frac{1}{e^{\frac{x^2}{2}}\cdot x}=0\\
\Lim_{x\to \infty}f(x)=\Lim_{x\to\infty}\frac{x^2}{e^{\frac{x^2}{2}}}=\Lim_{x\to \infty}\frac{2x}{e^{\frac{x^2}{2}}\cdot x}=\Lim_{x\to \infty}\frac{2}{e^{\frac{x^2}{2}}}=0\\
y=0\mbox{ asymptota}\)