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## Thursday, November 25, 2010

### Watched MIT 18.02 - lectures 9,10

The Second Derivative Test is a procedure for determining if a critical point is a maximum, a minimum, a saddle or a degenerate.

PROCEDURE:
- Calculate $f_x, f_y, f_{xx}, f_{xy}, f_{yy}$
- Calculate the critical points ;

Then for each critical point:
- Calculate $A = f_{xx}(x_0, y_0)$
- Calculate $B = f_{xy}(x_0, y_0)$
- Calculate $C = f_{yy}(x_0, y_0)$
- $AC-B^2$.

Apply the Second Derivative Test
If $AC-B^2 > 0$ and $A > 0$ minimum
If $AC-B^2 > 0$ and $A < 0$ maximum
If $AC-B^2 < 0$ saddle
If $AC-B^2 = 0$ degenerate

Click to enlarge

EXAMPLE:
( See Mathematica print )
$f(x,y)=e^{x^2-\frac{x^4}{4}-y^2}$

At $(0,0): \left\{AC-B^2, A\right\} = \left\{-4,2\right\}$

At $(-\sqrt{2},0): \left\{AC-B^2, A\right\} = \left\{8e^2,-4e\right\}$
Local maximum.

At $(\sqrt{2},0): \left\{AC-B^2, A\right\} = \left\{8e^2,-4e\right\}$
Local maximum.

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Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
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To create new theorems and proofs
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(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)