位置14的问题 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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BlackTom AI · Accepted Answer

The problem asks which equations you would solve with fsolve to locate the minimum of a differentiable function f(x, y). In optimization, a necessary condition for a (interior) minimum is that all first-order partial derivatives vanish, i.e., the gradient is zero. This means you must set both fx(x, y) = 0 and fy(x, y) = 0 and solve this system.

Option 1: fy(x, y) = 0 by itself. While setting fy = 0 is a valid component of the gradient condition, alone it does not determine a point where both partial derivatives are zero. It restricts you to a subset of the plane where the y-derivative is zero, but there is no guarantee fx is also zero there. Therefore this option is insufficient for finding a minimum.

Option 2: fx(x, y) = 0 and fy(x, y) = 0. This captures the full gradient-zero condition, requiring both partial derivatives to vanish simultaneously. This is exactly what you would solve with fsolve to locate potential stationary points that could be minima (subject to further second-derivative or Hessian analysis to confirm minimality in practice).

Option 3: fx(x, y) = 0 by itself. Similar to Option 1, setting only fx to zero does not ensure fy is zero, so it can miss stationary points or yield points that are not minima. It’s an incomplete condition.

Option 4: f(x, y) = 0. Setting the function value to zero is unrelated to finding where the gradient vanishes. A point can have f = 0 at many locations, including those where the gradient is nonzero, so this does not identify minima.

Option 5: You cannot find the minimum of f(x, y) using fsolve. This is incorrect in the context of unconstrained optimization: fsolve can find roots of a system of equations, and the gradient-zero system is exactly what you need to locate potential minima. While further checks are needed to confirm true minima, the statement that fsolve cannot be used for this purpose is false.

In summary, the correct approach is to solve the system fx(x, y) = 0 and fy(x, y) = 0 using fsolve, since both partial derivatives must vanish at a (potential) minimum. The other options either omit a necessary condition, rely on an irrelevant condition, or make an inaccurate broad claim about the method.

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