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- Find the minimum value of f (x, y) = xy subject to the constraint 5x - y = 4 in two ways: using Lagrange multipliers and setting y = 5x -4 in f(x,y).Use Lagrange multipliers to find the maximum and minimum values of f (x, y, z) = x2 + 2y2 + 3z2 subject tox + y + z = 1 and x - y + 2z = 2.find the maximum value that f(x, y, z) = x^2 + 2y - z^2 can have on the line of intersection of the planes 2x-y = 0 and y + z = 0 Can this problem be solved using lagrange multipliers?
- Solve max f( x,y , z) 3x2+y problem by means of Lagrange multipliers under the constraints of 4x-3y=9 and x2+z2=9. Explain what the values of Lagrange multipliers mean?Find the point on the planez = x + y + 1closest to the point P = (1, 0, 0). Hint: Minimize the square of thedistance.Use the method of Lagrange multipliers to show that for x, y ≥ 0, we have
- A function, z = ax + by, is to be optimized subject to the constraint, x2 + y2=1 where a and b are positive constants. Use Lagrange multipliers to show that this problem has only one solution in the positive quadrant (i.e. in the region x > 0, y > 0) and that the optimal value of z is √a2 +b2.Solve min x1-x2-2x3 problem by means of Lagrange multipliers under the constraints of x1+x2+x3=5 and x12+ x22 =4. Explain what the values of Lagrange multipliers mean?Use Lagrange multipliers to find the critical points and the relative extrema. f(x,y,z) = x² + y² + z² subject to the constraint 0 = x² - y² + 1
- Use Lagrange multipliers to find the minimum distance from the curve or surface Line: x − y = 4 to the point (0, 2).Find the point on the surface 4x+y-1=0 closest to the point (1,2,-3) using Larange Multipliers. I have only managed to determine that z=-3, but all the other x and y values I have solved for have not looked right compared to a graph.