Problems On Lagrange’s Method Of Undetermined Multipliers

Problems on Lagrange’s Method of Undetermined Multipliers:

1. Find the minimum value of x² y² z² subject to the conditions ax by cz = p.

Solutions: Let

But,

Hence the required minimum value of x² y² z²is

Thus the required minimum value is

2. The temperature T at any point (x,y,z) in space is T = 400 xyz². Find the heighest temperature at the surface of the unit sphere x² y² z² = 1

Solutions: Let

Taking the equality pairs, we get

The maximum temperature:

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