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## Homework Statement

Find the extrema of the function subject to the given constraint.

**f (x, y) = x**

x

^{2}+ 2y

^{2}= 3

## Homework Equations

det = f

_{xx}* f

_{yy}- (f

_{xy})

^{2}

If det > 0,

f

_{xx}< 0 [tex]\Rightarrow[/tex] MAXIMUM

f

_{xx}> 0 [tex]\Rightarrow[/tex] MINIMUM

If det < 0,

[tex]\Rightarrow[/tex] SADDLE POINT

If det = 0m

Inconclusive

## The Attempt at a Solution

I appear to be struggling with this maxima/minima stuff. Previous questions were

*just*correct. I had to look at the book's answers to figure a way there. I seem to be missing something...but here is my attempt:

Since x

^{2}+ 2y

^{2}= 3,

**x = [tex]\pm[/tex][tex]\sqrt{3 - 2y**

^{2}}[/tex]Therefore,

**f(x, y) = [tex]\pm[/tex][tex]\sqrt{3 - 2y**

^{2}}[/tex]At critical point,

**[tex]\nabla[/tex] f = (0, - [tex]\frac{2y}{\sqrt{3 - 2y**

^{2}}}[/tex] )**= (0, 0)**

Therefore, equating gives

**0 = 0**

&

**- [tex]\frac{2y}{\sqrt{3 - 2y**

^{2}}}[/tex] = 0So extrema points at

**([tex]\sqrt{3}[/tex], 0)**&

**(-[tex]\sqrt{3}[/tex], 0)**since y = 0.

To find max/min/saddle, find det.

**f**

f

det = f

det = 0 * f

det = 0

_{xx}= 0f

_{xy}= 0det = f

_{xx}* f_{yy}- (f_{xy})^{2}det = 0 * f

_{yy}- (0)^{2}det = 0

Which is inconclusive. The book has ([tex]\sqrt{3}[/tex], 0) as a maximum and (-[tex]\sqrt{3}[/tex], 0) as a minimum.

I don't know what I'm missing out here...

Thanks in advance.