Question

If 𝜙(x, y) = 0, show that the determinant

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fxx + λϕxx

fxy + λϕxy

ϕx

fxy + λϕxy

fyy + λϕyy

ϕy

ϕx

ϕy

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where 𝜆 is Lagrange’s multiplier, is positive, in case the function attains a maximum.

Answer 1

$\begin{vmatrix} f_{xx}+\lambda \phi_{xx} & f_{xy}+\lambda \phi_{xy} & \phi_x \\ f_{xy}+\lambda \phi_{xy} & f_{yy}+\lambda \phi_{yy} & \phi_y \\ \phi_x & \phi_y & 0 \end{vmatrix}$

$\det \begin{pmatrix} f_{xx}+\lambda \phi_{xx} & f_{xy}+\lambda \phi_{xy} & \phi_x\\ f_{xy}+\lambda \phi_{xy}& f_{yy}+\lambda \phi_{yy} & \phi_y \\ \phi_x&h&i\end{pmatrix}=f_{xx}+\lambda \phi_{xx} \cdot \det \begin{pmatrix} f_{yy}+\lambda \phi_{yy} &\phi_y\\ \phi_y&0\end{pmatrix}-f_{xy}+\lambda \phi_{xy}\cdot \det \begin{pmatrix}f_{xy}+\lambda&\phi_y\\ \phi_x &0\end{pmatrix}+\phi_x\cdot \det \begin{pmatrix}f_{xy}+\lambda& f_{yy}+\lambda \phi_{yy}\\ \phi_x&\phi_y\end{pmatrix}$

$[f_{xx}+\lambda \phi_{xx} \cdot (-\phi_y ^2)]-[f_{xy}+\lambda \phi_{xy}\cdot (-\phi_x\phi_y) ]+[\phi_x\cdot (\phi_{y})(f_{xy}+\lambda_{xy})-(\phi_{y})(f_{yy}+\lambda_{yy})]$