Study of the quartic y = x fourth minus 4x squared

Text

Study the function $y=x^{4}-4x^{2}$: domain, symmetry, intercepts, relative extrema and inflections.

Solution

Domain $\mathbb{R}$; even function. Intercepts: $x^{2}(x^{2}-4)=0\Rightarrow x=0,\ x=\pm2$. First derivative: $y'=4x(x^{2}-2)=0\Rightarrow x=0,\ x=\pm\sqrt2$. $x=0$ relative maximum $(0,0)$; $x=\pm\sqrt2$ minima $(\pm\sqrt2,-4)$. Second derivative: $y''=12x^{2}-8=0\Rightarrow x=\pm\sqrt{2/3}\approx\pm0.816$: two inflections. $\boxed{\text{max }(0,0);\ \text{min }(\pm\sqrt2,-4)}$