Answer to Question #273041 in Calculus for Aparna

Question #273041

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

1
Expert's answer
2021-11-30T10:04:18-0500

Let us locate the critical points of the function f(x)=4x416x2+17.f(x)= 4x^4-16x^2+17. It follows that f(x)=16x332x=16x(x22).f'(x)=16x^3-32x=16x(x^2-2). Then f(x)=0f'(x)=0 implies x=0x=0 or x=2x=-\sqrt{2} or x=2.x=\sqrt{2}. Therefore, the critical points are the following:

0,2,2.0,-\sqrt{2},\sqrt{2}.


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

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


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