Critical Points and the Second Derivative Test for a Multivariable Function

[sheepcountme]

Homework Statement

Find the critical points and use the second derivative test to decide if your critical points are local maxima, local minima, or saddle points.

f(x,y)=x⁴+y⁴+4xy

The Attempt at a Solution

so I took the gradient to get: <4x³+4y, 4y³+4x>

I know I need to set this equal to <0,0>..so,


4x³+4y=0 and 4y³+4x=0

but I'm stuck...I tried solving for y in the first one to get

y=-x³ and then plugging this into the next equation to get -x⁹+x=0

If I solve for x, I believe I get x=0 or x=1 and then plugging these into the first I get the points (0,0) and (1,-1)

Have I done this correctly?

And when we're talking about the second derivative test, is this the Hessian? And if so, I've gotten 144x²y²-12x²$$\lambda$$-12y²$$\lambda$$+$$\lambda$$²-16=0

which seems awfully messy to be able to determine if the point is a maxima, etc.

 

[Dick]

That seems pretty ok, but what's wrong with (-1,1) as a critical point? I don't think you found all of the roots. And what are those lambdas doing the Hessian? Isn't it just the determinant of the matrix of second derivatives?

 

[HallsofIvy]

Homework Statement

Find the critical points and use the second derivative test to decide if your critical points are local maxima, local minima, or saddle points.

f(x,y)=x⁴+y⁴+4xy

The Attempt at a Solution

so I took the gradient to get: <4x³+4y, 4y³+4x>

I know I need to set this equal to <0,0>..so,


4x³+4y=0 and 4y³+4x=0

but I'm stuck...I tried solving for y in the first one to get

y=-x³ and then plugging this into the next equation to get -x⁹+x=0

If I solve for x, I believe I get x=0 or x=1 and then plugging these into the first I get the points (0,0) and (1,-1)

$x^9- x= x(x^8- 1)= 0$
Yes, x= 0 is a root. But $x^8- 1= 0$ has two real roots.

Have I done this correctly?

And when we're talking about the second derivative test, is this the Hessian? And if so, I've gotten 144x²y²-12x²$$\lambda$$-12y²$$\lambda$$+$$\lambda$$²-16=0

which seems awfully messy to be able to determine if the point is a maxima, etc.

Look at
$$\left|\begin{array}{cc}f_{xx} & f_{xy} \\ f_{xy} & f_{yy}\end{array}\right|= \left|\begin{array}{cc}12x^3 & 4 \\ 4 & 12 y^3\end{array}\right|$$
Put in the values of x and y before evaluating the determinant and it is not at all complicated!

 

[sheepcountme]

Ah! I missed (-1,1), thank you!

And I got eigenvalues unnecessarily mixed up in all this (where the lamdas came from), thanks!