Does f(x,y) = yx² – 2x2 – y² have any local extrema or saddle points? If so, identify them; if not, say why not. It may be helpful in this problem to recall the Hessian of a function f(x,y): |fxx fxy| fry fyyl (the vertical bars denote a determinant)
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- Compute the Wransky determinant (Wranskian) of the functions y1, y2, y3 defined below with constants a, b, c y1(x) = 8a + 7x, y2(x) = xln(bx), y3(x) = cx^2Find all stationary points of the function f(x,y)= 3x2 -6xy+2y3, and use Hessian Matrix and the determinant to classify each as a local maximum, a local minimum, or a saddle point.Find the extreme value(s) for the following function and determine local (global) maximum or minimum (Check the second-order conditions. Use Hessian Determinant) z=e2x + e-y + ew2 -(2x + 2ew - y)
- This is a 5 part problem and have to do with the region D, which is bounded by xy = 1, xy = 9, xy^2 = 1, and xy^2 = 25 in the first quadrant of the xy plane. a) Graph the region D. b) Using the non-linear change of variables u = xy and v = xy^2, find x and y as functions of u and v. c) Find the determinant of the Jacobian for this change of variables. d) Using the change of variables, set up a double integral for calculating the area of the region D. e) Evaluate the double integral and compute the area of the region D.Compute the discriminant D(x, y) of ƒ(x, y) = x2y2.Why Wronskian determinant W(a) = |X(a)| is nonzero?
- Evaluate the determinants D,Dx,Dy, and Dz.Let A(t) be a n × n matrix whose coefficients are continuous functions of t on the interval (α, β). Let t0 ∈ (α, β) and asssume that y(t) is a solution of the initial value problem x' = A(t)x, x(t0) = 0. Show that y(t) = 0 for all t ∈ (α, β).Let X = X(t) be the response of the linear dynamical system x' = ?x − ?y y' = ?x + ?y that satisfies the initial condition X(0) = X0. Determine conditions on the real constants ? and ? that will ensure lim t→∞ X(t) = (0, 0). (Enter your answer as a comma-separated list of equations and inequalities.)