# Questions tagged [hessian-matrix]

The Hessian matrix of function is used to second derivative test when $f$ has a critical point $x$. If the Hessian is positive definite at $x$, then $f$ attains a local minimum at $x$. If the Hessian is negative definite at $x$, then $f$ attains a local maximum at $x$. If the Hessian has both positive and negative eigenvalues then $x$ is a saddle point for $f$.

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### Hessian determinant of a function

I have a function with two variables ($f(x,y)$). Hessian determinant of the function is always equal to zero: $f_{xx}f_{yy}-f_{xy}f_{yx}=0 ~~\forall x,y$ and both $f_{xx}$ and $f_{yy}$ are always ...
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### Proving there is $x\in\Bbb R^n,\|x\|=1$ s. t. $f(x)>f(y),\forall y\in B(0,1),$ where $f\in C^2(A,\Bbb R)$

Let $A\subseteq\Bbb R^n$ be an open set which contains the closed unit ball $\overline{B(0,1)}$ and let $f\in C^2(A,\Bbb R).$ Suppose the Hessian $H_f(c)$ is positive definite $\forall c\in B(0,1).$ ...
If I have a Hessian matrix $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ might someone help me understand why this is not positive semidefinite? My understanding that if for ...