Chapter 4: Partial Differentiation
Section 4.8: Unconstrained Optimization
Find the minimum distance between the point 1,−2,3 and the plane 5 x+3 y+2 z=7.
Take as the objective function fx,y,z=x−12+y+22+z−32, the square of the distance of the arbitrary point x,y,z from the point 1,−2,3, and let the constraint be the plane described by gx,y,z≡5 x+3 y+2 z−7=0.
The Lagrange multiplier method requires solving the equations in ∇f=λ ∇g, along with g=0, for x,y,z,λ. These equations are
and have solution x,y,z=24/19,−35/19,59/19 ≐ 1.26,−1.84,3.11 with λ=2/19. The minimum distance between the point 1,−2,3 and the given plane is then 2/19≐0.324.
Maple Solution - Interactive
Context Panel: Assign to a Name≻f
x−12+y+22+z−32→assign to a namef
Context Panel: Assign to a Name≻g
5 x+3 y+2 z−7→assign to a nameg
Context Panel: Assign to a Name≻F
f−λ g→assign to a nameF
Form and solve the equations of the Lagrange multiplier method
Calculus palette: Partial-differentiation operator
Press the Enter key.
Context Panel: Solve≻Solve
Context Panel: Assign to a Name≻S
∂∂ x F=0,∂∂ y F=0,∂∂ z F=0,g=0
→assign to a name
Obtain the minimum distance between the given point and the given plane
Expression palette: Evaluation template
Evaluate and Display Inline
Context Panel: Approximate≻5 (digits)
fx=a|f(x)S = 119⁢38→at 5 digits0.32444
A numeric solution can be obtained interactively as follows.
Write the sequence of objective function (here, f) and constraint equation; press the Enter key.
Context Panel: Optimization≻Minimize (local)
The first number in the return is the minimum distance, that is, the minimum of f subject to the constraint g=0. The list of equations that follows give the coordinates of the point on the constraint plane where this minimizing point lies.
Maple Solution - Coded
Define the objective function f.
Define the constraint function g.
g≔5 x+3 y+2 z−7:
Implement the Lagrange multiplier technique via the LagrangeMultipliers command
Alternatively, use the Equate and eval commands to evaluate f at the solution S.
evalf,Equatex,y,z,S = 219
Find the minimum distance between the given point and the given plane
Apply the evalf command.
evalf2/19 = 0.3244428423
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