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I know that what the formula is used to determine a point whether it has maximum, minimum, or saddle points; but, i do not understand how it is formulated.

$$ H = f_{xx}​(x_0​,y_0​)f_{yy}​(x_0​, y_0​) − f_{xy}​(x_0​, y_0​)^2 $$

The $f_{xx}​(x_0​,y_0​)$ and $f_{yy}​(x_0​,y_0​)$ are used to determine the concavity in x and y direction respectively, but why it is multiplied by each other. I also do not understand why it is subtracted by $f_{xy}​(x_0​,y_0​)^2$ and why it has a power of two?

Greg Nisbet
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dave
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1 Answers1

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The single-variable condition $f''(x_0)>0$ is replaced by $Hf(x_0,y_0)>0$, where $Hf$ stands for the Hessian of $f$. But this is a matrix, so this ''$>0$'' has to be read carefully: it means that the matrix $Hf(x_0,y_0)$ is positive definite. Since this matrix has order $2$, positive-definiteness is equivalent to the conditions $f_{xx}(x_0,y_0)>0$ and $\det Hf(x_0,y_0)>0$ holding together. This is what you have there. For higher dimensions, look up "Sylvester's Criterion".

Ivo Terek
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  • You can figure some of this out for yourself by examining the quadratic function $Q(x,y)=Ax^2+2Bxy+Cy^2$ and seeing that $AC-B^2$ shows up when you complete the square. – Ted Shifrin Feb 05 '21 at 05:54