The 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 + y

To find the interval for k where (0,0) is a critical point of the function f(x, y) = x² + kxy + y, we first 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: fx = 2x + ky The partial derivative of f with respect to y is: fy = kx + 1

Setting these equal to zero gives us:

2x + ky = 0 kx + 1 = 0

Substituting (0,0) into these equations gives us:

20 + k0 = 0 k*0 + 1 = 0

Solving these equations gives us k = -1.

However, we need to check the second order conditions to ensure that (0,0) is a critical point. The second order conditions are given by the determinant of the Hessian matrix, which is the matrix of second derivatives.

The second derivatives are:

fxx = 2 fxy = k fyx = k fyy = 0

The Hessian matrix is therefore:

H = | 2 k | | k 0 |

The determinant of the Hessian matrix is (20) - (kk) = -k².

For (0,0) to be a critical point, the determinant of the Hessian matrix must be greater than zero. This gives us the inequality -k² > 0.

However, this inequality has no solution for real numbers k, because the square of a real number is always non-negative. Therefore, there is no interval (a, b) of k ∈ R for which (0,0) is a critical point of the function f(x, y) = x² + kxy + y.