91Ó°ÊÓ

We have given the following function :-

$g\left(x,y\right)=-\left(x^4+y^4\right)$.

We have to show that $g\left(x,y\right)$has a stationary point at the origin $\left(0,0\right)$.

The given function is :-

$g\left(x,y\right)=-\left(x^4+y^4\right)$.

Now partially differentiate this function with respect to $x$and $y$, then we have :-

$g_x=-4x^3$

and

$g_y=-4y^3$.

Now put $g_x=0=g_y$, then we have :-

$$\begin{gathered} -4x^3=0 \\ ⇒x^3=0 \\ ⇒x=0 \end{gathered}$$

and

$$\begin{gathered} -4y^3=0 \\ ⇒y^3=0 \\ ⇒y=0 \end{gathered}$$

That is the stationary point is $\left(0,0\right)$.

Hence it is proved that the function $g\left(x,y\right)=-\left(x^4+y^4\right)$has stationary point at origin.

We find that :-

$g_x=-4x^3$and $g_y=-4y^3$.

Now find second derivatives :-

$g_{xx}=-12x^2$. Then :-

$g_{xx}(0,0)=0$

and

$g_{yy}=-12y^2$. Then :-

$g_{yy}(0,0)=0$

and

$g_{xy}=0$

We know that $det\left(H_g\right)$is defined as $det\left(H_g\right)=g_{xx}{g}_{yy}-{\left(g_{xy}\right)}^2$.

Then :-

$$\begin{gathered} det\left(H_g\left(0,0\right)\right)=\left(0×0\right)-{\left(0\right)}^2 \\ ⇒det\left(H_g\left(0,0\right)\right)=0 \end{gathered}$$

Hence proved.

The given function is :-

$g\left(x,y\right)=-\left(x^4+y^4\right)$

We know that :-

$$\begin{gathered} x^4,y^4â‰\yen 0 \\ ⇒x^4+y^4â‰\yen 0 \\ ⇒-\left(x^4+y^4\right)≤0 \end{gathered}$$

So we can say that the value of the function is always non positive.

That is $g\left(x,y\right)≤0$.

At the origin $\left(0,0\right)$, then value of the function is :-

$$\begin{gathered} g\left(0,0\right)=-\left(0+0\right) \\ ⇒g\left(0,0\right)=0 \end{gathered}$$

This is the possible maximum value of the function.

So we can conclude that the given function $g$is absolute maximum at origin.