LaGrange’s method of multipliers with one subsidiary condition
Procedure
Step 1:
Auxilliary function is formed by
Step 2:
Form equations
Fx=0 ,Fy=0 and Fz=0 where
Fx is the partial derivative of ‘F’ with respect to ‘x’
Fy is the partial derivative of ‘F’ with respect to ‘y’ and
Fz is the partial derivative of ‘F’ with respect to ‘z’.
Step 3:
Solve for (x,y,z) and ‘λ’,the values of u(x,y,z)are the Stationary values.
LaGrange’s Method Problems
- Find the minimum value of x^2+y^2+z^2 subject to the condition xy+yz+zx=3a^2
- Discuss maximum or minimum for x^3+y^3-3xy
- Find the maxima and minima values of the function f (x, y) =3x+4y on the circle x^2+y^2=1 using method of Lagrange’s multipliers
- Find the maximum and minimum values of x^2+y^2 subject to the condition 2x^2+3xy+2y^2=1
- If x, y, z are the angles of a triangle, find the maximum value of sin x sin y sin z
- A rectangular box open at the top is to have a volume of 32 cubic feet. Find its dimensions if the total surface area is minimum
- Show that the rectangular box of maximum value and a given surface area is a cube
- Find the volume of the largest rectangular parallelepiped that can be inscribed in the ellipsoid x^2/a^2 +y^2/b^2 +z^2/c^2 =1
- Find the dimensions of the rectangular box open at the top of maximum capacity whose surface is 432 sq.cm