Question 1: Find the values of parameter m so that the graph of the function: \(y = {x^4} – 2{m^2}{x^2} + {m^4} + 1\) has three extreme point. At the same time, those three extreme points together with the origin O form an inscribed quadrilateral

Then the 3 extreme points are

\(A\left( {0;{m^4} + 1} \right),B\left( { – m;1} \right),C\left( {m;1} \right)\)

Let I be the center of the circumcircle (if any) of quadrilateral ABOC.

Due to symmetry, we have

A,O,I in a straight line

⇒ AO is the diameter of the circumcircle (if any) of quadrilateral ABOC

\(\begin{array}{l}

AB \bot OB \Leftrightarrow \overrightarrow {AB} .\overrightarrow {OB} = 0\\

\Leftrightarrow {m^2} – {m^4} = 0 \Leftrightarrow \left[\begin{array}{l}[\begin{array}{l}

m = 0\\

m = \pm 1

\end{array} \right.

\end{array}\)

Combination of conditions first ( satisfy)

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