Expert: Paul Klarreich - 6/8/2006

Question
Hi Tony here
Not studying at the moment but need this in a compression program I am trying out just for the hell of it?

can you check that I have done the correct thing to find the maximums & minimums
of the following equation

f(x,y) = A + By + Cx + Dxy + E(y^2) + N(x^2)

where A,B,C,D,E,N are constants &
y,x variables

this is what I have forgotten, differentiating with respect to more than 1 variable but here
goes

f'(x,y) = B + C + D + 2Ey + 2Nx

maximums, minimums & points of inflection occur when

2Ey + 2Nx = -B - C - D

Answer
Hi, Tony,

Subject:  3D differentiation
Question:  Hi Tony here
Not studying at the moment but need this in a compression program I am trying out just for the hell of it?

can you check that I have done the correct thing to find the maximums & minimums of the following equation

>> you mean function of two variables.

f(x,y) = A + By + Cx + Dxy + E(y^2) + N(x^2)

where A,B,C,D,E,N are constants & y,x variables

this is what I have forgotten, differentiating with respect to more than 1 variable but here goes

f'(x,y) = B + C + D + 2Ey + 2Nx

maximums, minimums & points of inflection occur when

2Ey + 2Nx = -B - C - D
---------------------------------

Sorry, but that does not look right.  You should compute the partial derivatives.  (If you have never heard of these things, it's back to school for the third semester of calculus.)

I have to write 'd' for the partial derivative symbol here -- the interface is rather crude.  

df/dx = C + Dy + 2Nx
df/dy = B + Dx + 2yE

Now you will have your max-min at a point (x,y) that makes both of those derivatives equal to zero.  (You have simultaneous equations to solve.)  Having found such a point, you can investigate further and find that the point will be:

A. A maximum point if it is a max in both x- and y-directions.
B. A minimum point if it is a min in both x- and y-directions.
C. A saddle point if it is a max in one and a min in the other direction.

You can use second derivatives to determine that.