Given the functions

\( f(x,y) = \frac{x}{2} e^{ - \big(\frac{x^2 + y^2}{2} \big)} \)

\( g(z) = \log_2 (\frac{z}{5}) \)

and the matrices

A = \( \begin{bmatrix} \\ {{\partial f(x,y) \over \partial x} \bigg|_{(0,0)}} & {{\partial^2 f(x,y) \over \partial x \partial y} \bigg|_{(1,1)}} \\ \\ {{\partial^2 f(x,y) \over \partial y \partial x} \bigg|_{(0,0)}} & {{\partial f(x,y) \over \partial y} \bigg|_{(1,1)}} \\ \\ \end{bmatrix} \)

B = \( \begin{bmatrix} g'(1) & g'(2) \\ g''(1) & g''(2) \\ \end{bmatrix} \)

find \( k = \ln (2) . \frac{det B}{det A} \)

**Notations:**

- \( {{\partial f(x,y) \over \partial x} \bigg|_{(a,b)}} \) denotes the first-order partial derivative of \(f(x,y)\)

with respect to \( x \) at the point \((a,b,f(a,b))\)

- \( {{\partial f(x,y) \over \partial y} \bigg|_{(a,b)}} \) denotes the first-order partial derivative of \(f(x,y)\)

with respect to \( y \) at the point \((a,b,f(a,b))\)

- \( {{\partial^2 f(x,y) \over \partial x \partial y} \bigg|_{(a,b)}} \) denotes the second-order partial derivative of \(f(x,y)\)

with respect to \( y \) and \( x \) at the point \((a,b,f(a,b))\)

- \( {{\partial^2 f(x,y) \over \partial y \partial x} \bigg|_{(0,0)}} \) denotes the second-order partial derivative of \(f(x,y)\)

with respect to \( x \) and \( y \) at the point \((a,b,f(a,b))\)

\( g'(a) \) denotes the first derivative of \( g(z) \) at the point \( (a, g(a)) \)

\( g''(a) \) denotes the second derivative of \( g(z) \) at the point \( (a, g(a)) \)

\( \log_2 (\cdot) \) denotes the binary logarithm of \( (\cdot) \)

\( \ln (\cdot) \) denotes the natural logarithm of \( (\cdot) \)

\( det A \) denotes the determinant of matrix A

\( det B \) denotes the determinant of matrix B

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