4. Znajdx i określ ekstrema funkcji

[Januszgolenia]

\(y=x^2-arctg(2x^2)\)

[eresh]

\(y=x^2-arctg(2x^2)\)

\(f'(x)=2x-\frac{4x}{1+4x^4}\\
f'(x)=\frac{2x+8x^5-4x}{1+4x^4}\\
f'(x)=\frac{8x^5-2x}{1+4x^4}\\
f'(x)=\frac{2x(4x^4-1)}{1+4x^4}\\
f'(x)=\frac{2x(2x^2+1)(2x^2-1)}{1+4x^4}\\
f'(x)>0\iff x\in (-\frac{\sqrt{2}}{2},0)\cup (\frac{\sqrt{2}}{2},\infty)\\
f'(x)<0\iff x\in (-\infty, -\frac{\sqrt{2}}{2})\cup (0,\frac{\sqrt{2}}{2})\\
f_{max}=f(0)\\
f_{min}=f(-\frac{\sqrt{2}}{2})\\
f_{min}=f(\frac{\sqrt{2}}{2})
\)