# constrained optimisation problem

#### lin.13579

(a) Write down the objective function and the constraint equation for aconstrained optimisation problem which is to minimize the total surface area of a rectangular box with the constraint that its volume is fixed to be 1. (Hint: Let the box have length x, width y and height z.)

(b) Solve the problem using the method of Lagrange multipliers.

Is anyone teach me how to do this question? thanks

#### Prove It

MHF Helper
(a) Write down the objective function and the constraint equation for aconstrained optimisation problem which is to minimize the total surface area of a rectangular box with the constraint that its volume is fixed to be 1. (Hint: Let the box have length x, width y and height z.)

(b) Solve the problem using the method of Lagrange multipliers.

Is anyone teach me how to do this question? thanks
To start you off, you are required to:

$$\displaystyle \textrm{Minimise }\,2xy + 2xz + 2yz$$

$$\displaystyle \textrm{subject to }\,xyz = 1$$.

#### HallsofIvy

MHF Helper
To minimize F(x,y,z), subject to G(x,y,z)= constant, use $$\displaystyle \nabla F= \lambda \nabla G$$ where $$\displaystyle \lambda$$ is the "Lagrange multiplier". Setting components equal gives three equations in the four unknowns, x, y, z, and $$\displaystyle \lambda$$. The constraint G(x,y,z)= constant is a fourth equation.

Since the value of $$\displaystyle \lambda$$ is not part of the solution, I find that dividing one equation by another to eliminate $$\displaystyle \lambda$$ is often a good first step.