Question at position 14 You want to minmize a differentiable function f(x,y) but decide to use scipy.optimize.fsolve instead of scipy.optimize.minimize. Which of the following equations, or systems of equations, would you need to solve using fsolve to find the minimum of f(x,y)? Let fx(x,y) and fy(x,y) denote the partial derivatives of f with respect to x and y, respectively. fy(x,y)=0fx(x,y)=0 and fy(x,y)=0fx(x,y)=0f(x,y)=0You cannot find the minimum of f(x,y) using fsolve.

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The problem asks which equations you would need to solve with fsolve in order to locate the minimum of a differentiable function f(x, y). This hinges on finding stationary points where the gradient is zero.

Option A: "f_y(x, y) = 0". This only sets the partial derivative with respect to y to zero. It ignores the x-direction, so you could miss critical points where f_x ≠ 0. Therefore it cannot uniquely identify minima.

Option B: "f_x(x, y) = 0 and f_y(x, y) = 0". This enforces both partial derivatives to be zero, i.e., the gradient ∇f = (f_x, f_y) equals (0, 0). These are exactly the first-order optimality conditions for stationary points. In optimization terminology, these are the equations you solve with fsolve to locate potential minima (subject to further checks like the Hessian for confirming a minimum).

Option C: "f_x(x, y) = 0". Like A, this only constrains the x-direction and ignores the y-direction. Any point where f_y ≠ 0 could still be a saddle or non-stationary point, so this cannot identify minima on its own.

Option D: "f(x, y) = 0". Setting the function value to zero describes a level curve (or surface) of f, not the condition for a minimum. Minimization requires the gradient to be zero, not that the function itself equals zero.

Option E: "You cannot find the minimum of f(x, y) using fsolve.". This is incorrect in general: fsolve can locate stationary points by solving ∇f = 0, and while it doesn’t guarantee identifying a minimum (could be a saddle or maximum), it is commonly used to find candidate minima via the gradient conditions.

Putting it together, only Option B correctly specifies the necessary and sufficient first-order conditions to locate a minimum candidate via fsolve, with the others lacking essential components or mischaracterizing what needs to be solved.

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