Nonlinear ProgrammingThe largest interval (a, b) of k ∈ R,for which the point (0, 0) is the critical point of the function f(x, y) = x2 + kxy + yans.

To find the interval (a, b) of k ∈ R for which the point (0, 0) is the critical point of the function f(x, y) = x² + kxy + y, we need to find the partial derivatives of the function and set them equal to zero.

The partial derivative of f with respect to x is 2x + ky. The partial derivative of f with respect to y is kx + 1.

Setting these equal to zero gives us the equations:

2x + ky = 0 kx + 1 = 0

We are looking for the values of k for which (0,0) is a solution to these equations. Substituting x = 0 and y = 0 into the equations gives us:

0 = 0 1 = 0

The second equation is a contradiction, which means that there are no values of k for which (0,0) is a critical point of the function. Therefore, the interval (a, b) of k ∈ R for which the point (0, 0) is the critical point of the function f(x, y) = x² + kxy + y is empty.