- #1
RyanV
- 12
- 0
Homework Statement
Find the extrema of the function subject to the given constraint.
f (x, y) = x
x2 + 2y2 = 3
Homework Equations
det = fxx * fyy - (fxy)2
If det > 0,
fxx < 0 [tex]\Rightarrow[/tex] MAXIMUM
fxx > 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 x2 + 2y2 = 3,
x = [tex]\pm[/tex][tex]\sqrt{3 - 2y2}[/tex]
Therefore,
f(x, y) = [tex]\pm[/tex][tex]\sqrt{3 - 2y2}[/tex]
At critical point,
[tex]\nabla[/tex] f = (0, - [tex]\frac{2y}{\sqrt{3 - 2y2}}[/tex] )
= (0, 0)
Therefore, equating gives
0 = 0
&
- [tex]\frac{2y}{\sqrt{3 - 2y2}}[/tex] = 0
So extrema points at ([tex]\sqrt{3}[/tex], 0) & (-[tex]\sqrt{3}[/tex], 0) since y = 0.
To find max/min/saddle, find det.
fxx = 0
fxy = 0
det = fxx * fyy - (fxy)2
det = 0 * fyy - (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.