Exercise · ZW-HESSIAN-MATRIX-008 · ★★☆ Intermediate
For \(k>0\), which value makes the Hessian of \(f(x,y)=x^2+kxy+y^2\) singular?
Choose one option.
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- D. 2
Explanation
\(H=\begin{pmatrix}2&k\\k&2\end{pmatrix}\), \(\det H=4-k^2=0\) gives \(k=\pm2\); with \(k>0\), \(k=2\).
Review the concept: Hessian matrix
The Hessian matrix \(H\) of a function \(f\) of two variables is the \(2\times2\) matrix of its second-order partial derivatives at a point.
\[H=\begin{pmatrix}f_{xx}&f_{xy}\\ f_{yx}&f_{yy}\end{pmatrix}\]
More practice problems: hessian matrix
- ZW-HESSIAN-MATRIX-001 Compute the Hessian matrix of f(x,y)=x^2+3xy+y^2.
- ZW-HESSIAN-MATRIX-002 Compute the Hessian of f(x,y)=x^3y at (1,1).
- ZW-HESSIAN-MATRIX-003 Compute det H for f(x,y)=x^2+xy+y^2.
- ZW-HESSIAN-MATRIX-004 Compute the Hessian of f(x,y)=e^xy at (0,0).
- ZW-HESSIAN-MATRIX-005 Compute the Hessian of f(x,y)=xln y at (1,1).
- ZW-HESSIAN-MATRIX-006 If f has continuous second-order partial derivatives, what always holds for its Hessian H?
- ZW-HESSIAN-MATRIX-007 Compute the trace of the Hessian of f(x,y)=x^3+y^3-3xy at (1,1).
- ZW-HESSIAN-MATRIX-009 Compute det H at (1,1) for f(x,y)=x^2y^2.