# 극대,극소,안장점(3D)

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def illustrate_test(f, fvals=(-5,5), tvals=(-3,3)): """ Return a table of critical points, eigenvalues, and eigenvectors, plus a 3D graph illustrating the curvature at each critical point. INPUTS: f -- a function. Usually this should be a callable function, defined like f(x,y)=x^2+y^2, for example. fvals (default (-5,5)) -- minimum and maximum values for the function plot tvals (default (-3,3)) -- minimum and maximum values for the parameter in the curves at each critical point """ line_thickness=3 surface_color='blue' line_color = ['red','blue'] x,y,t=var('x,y,t') tmin, tmax = tvals fmin, fmax = fvals f_parametric = vector([x,y,f(x,y)]) picture = plot3d(f,(x,fmin,fmax),(y,fmin,fmax), opacity=0.3, surface_color=surface_color) hessian=f(x,y).hessian() solns=solve([eqn(x,y)==0 for eqn in f.gradient()],[x,y], solution_dict=True) table=[['Color','Critical Point', 'Eigenvalue', 'Eigenvector']] pictures=[] i=0 for solution in [s for s in solns if s[x] in RR and s[y] in RR]: soln = dict([(key,RDF(val)) for key,val in solution.items()]) critical_point = f_parametric(soln) hessian_at_point = hessian(soln).change_ring(RDF) evals,evecs=hessian_at_point.eigenmatrix_right() evals = evals.diagonal() picture += point3d(critical_point.list(), color='green', size=10) for eval,evec in zip(evals,evecs.columns()): direction_plot = evec.plot(width=line_thickness,color=line_color[i%2]).plot3d().translate(critical_point) curve_formula = f_parametric(x=soln[x]+t*evec[0],y=soln[y]+t*evec[1]) curve = parametric_plot(curve_formula, (t,tmin,tmax),color=line_color[i%2],thickness=line_thickness) picture+=curve+direction_plot pictures += [curve] table.append(["<span style='color: %(c)s'>%(c)s</span>"%{'c':line_color[i%2]},critical_point, eval, (evec)]) i+=1 return table, picture var('x,y') @interact def _(f=selector([-x^2+y^2, x*y*exp((-x^2-y^2)/3), x^3-3*x+y^2-4*y], label="$f(x,y)=$")): f(x,y)=f table, graph=illustrate_test(f) html.table(table,header=True) show(graph)

 $f(x,y)=$ -x^2 + y^2 x*y*e^(-1/3*x^2 - 1/3*y^2) x^3 + y^2 - 3*x - 4*y

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