Answer to Question #273041 in Calculus for Aparna

Question

Locate the critical points and identity the stationary points of 4x^4-16x^2+17

Answer 1

Let us locate the critical points of the function $f(x)= 4x^4-16x^2+17.$ It follows that $f'(x)=16x^3-32x=16x(x^2-2).$ Then $f'(x)=0$ implies $x=0$ or $x=-\sqrt{2}$ or $x=\sqrt{2}.$ Therefore, the critical points are the following:

$0,-\sqrt{2},\sqrt{2}.$

Let us identity the type of stationary points. It follows that $f'(x)<0$ on the intervals $(-\infty,-\sqrt{2})$ and $(0,\sqrt{2})$. Also $f'(x)>0$ on the intervals $(-\sqrt{2},0)$ and $(\sqrt{2},+\infty).$

We conclude that $-\sqrt{2}$ and $\sqrt{2}$ are the points of minimum, and the point of maximum.