<s>[INST] What are the minimum and maximum of $f(x) = x^6 - 5x^4 + 5x^2 - 1$? I am stuck in obtaining the critical points:
$f'(x) = 6x^5 - 20x^3 + 10x$
$f''(x) = 30x^4 - 60x^2 + 10$
For critical points, we have: $f'(x) = 0$
$2x(3x^4 - 10x^2 + 5) = 0$
Now, how should i solve them to get the critical points. And what should the minima and maxima of the given function be at the end?
 [/INST] $2x(3x^4 - 10x^2 + 5) = 0$
$x_1=0$
$3x^4 - 10x^2 + 5 = 0$
Let $y=x^2$ then $3y^2 - 10y + 5 = 0$ and $y_{1,2}=\frac{5\pm\sqrt{10}}{3}$
and
$x_2=-\sqrt{\frac{5-\sqrt{10}}{3}}$, $x_3=\sqrt{\frac{5-\sqrt{10}}{3}}$, $x_4=-\sqrt{\frac{5+\sqrt{10}}{3}}$, $x_5=\sqrt{\frac{5+\sqrt{10}}{3}}$
So the maximum is $\infty$
Function is minimum when $x=-\sqrt{\frac{5+\sqrt{10}}{3}}$ or $x= \sqrt{\frac{5+\sqrt{10}}{3}}$
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