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elfy
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Homework Statement
Find the stationary points of the function:
z = ax^2 + by^2 + c
For each of the following sub-cases, identify any maxima and minima.
i) a > 0, b > 0
ii) a < 0, b < 0
iii) a and b of opposite signs.
Homework Equations
z = ax^2 + by^2 + c
The Attempt at a Solution
Z'(x) = 2ax = 0
Z'(y) = 2by = 0
Z'(x) = Z'(y) => 2ax = 2by
Solving for y: Y = ax/b
Solving for x: X = by/a
Z''(xx) = 2a
Z''(yy) = 2b
Z''(xy) = 0
Checking the condition [Z''(xx) * Z''(yy)] - (Z''(xy))^2 --> (2a*2b) - (0)^2 = 4ab.
For i) where a and b are > 0 --> 4ab > 0 and 2a > 0 ---> Minimum
For ii) where a and b are < 0 --> 4ab > 0 and 2a < 0 ---> Maximum
For iii) where a and b are opposite --> 4ab < 0 --> Saddle Point
Have I done this correctly? Because I can help thinking that I have forgotten something. I would really appreciate any help and input with regards to my attempted solution! - Thanks!