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.[0, 1](−∞, ∞)[-1,-1][1,0]This Question Is
Question
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.[0, 1](−∞, ∞)[-1,-1][1,0]This Question Is
Solution
The function given is f(x, y) = x^2 + kxy + y.
A critical point of a function occurs where the first derivative of the function is zero or undefined.
The partial derivatives of the function f(x, y) are:
∂f/∂x = 2x + ky ∂f/∂y = kx + 1
Setting these equal to zero gives:
2x + ky = 0 kx + 1 = 0
For the point (0,0) to be a critical point, both these equations must be satisfied.
Substituting x = 0 and y = 0 into the equations gives:
0 = 0 (which is always true) 1 = 0 (which is never true)
Therefore, there is no value of k for which (0,0) is a critical point of the function f(x, y) = x^2 + kxy + y.
So, the correct answer is none of the options provided.
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