How to find the maximum value of the function f ( x , y , z ) = x + 2 y - 3 z subject to the constraint z = 4 x 2 + y 2 ?

Question

How to find the maximum value of the function             f              (        x        ,        y        ,        z        )              =    x    +    2    y    -    3    z   subject to the constraint       z    =    4          x      2        +          y      2      ?

Jakob Howell · Accepted Answer

Forming the lagrangian      L          (      x      ,      y      ,      z      ,      λ      )        =          f              (        x        ,        y        ,        z        )              +    λ          g              (        x        ,        y        ,        z        )            with             g              (        x        ,        y        ,        z        )              =    4          x      2        +          y      2        -    z    =    0  calculating and solving for the stationary points            g              (        x        ,        y        ,        z        )              =    4          x      2        +          y      2        -    z    =    0  or            {                                    1            +            8            λ            z            =            0                                                2            +            2            λ            y            =            0                                                3            +            λ            =            0                                                4                          x              2                        +                          y              2                        -            z            =            0                              and solving for             (      x      ,      y      ,      z      ,      λ      )       we obtain            {                                    x            =                          1              24                                                            y            =                          1              3                                                            z            =                          17              144                                                            λ            =            -            3                              and the maximum value is            17      48      NOTE: To qualify the stationary point it is necessary to form            (      f      ∘      g      )              (      x      ,      y      )        =    x    +    2    y    -    3          (      4              x        2            +              y        2            )      and then calculate       H    =          ∇      2              (      f      ∘      g      )        =          (                                    -            24                                0                                                0                                -            6                              )      As we can observe,       H   has negative eigenvalues indicating that the found solution represents a maximum.

Guadara0c2x · Accepted Answer

Given:             f              (        x        ,        y        ,        z        )              =    x    +    2    y    -    3    z   and the constraint function             g              (        x        ,        y        ,        z        )              =    4          x      2        +          y      2        -    z    =    0  The Lagrange function is:      L          (      x      ,      y      ,      z      ,      λ      )        =          f              (        x        ,        y        ,        z        )              +    λ          g              (        x        ,        y        ,        z        )                  L          (      x      ,      y      ,      z      ,      λ      )        =    x    +    2    y    -    3    z    +    4    λ          x      2        +    λ          y      2        -    λ    z  We compute the partial derivatives:                    ∂        L                  (          x          ,          y          ,          z          ,          λ          )                            ∂        x              =    1    +    8    λ    x                      ∂        L                  (          x          ,          y          ,          z          ,          λ          )                            ∂        y              =    2    +    2    λ    y                      ∂        L                  (          x          ,          y          ,          z          ,          λ          )                            ∂        z              =    -    3    -    λ                      ∂        L                  (          x          ,          y          ,          z          ,          λ          )                            ∂        λ              =    4          x      2        +          y      2        -    z  These 4 derivatives should be set to zero before they are solved as a system of equations:      0    =    1    +    8    λ    x                 [1]            0    =    2    +    2    λ    y                 [2]            0    =    -    3    -    λ                 [3]            0    =    4          x      2        +          y      2        -    z                 [4]      Solve equation [3] for       λ  :      λ    =    -    3  Substitute -3 for       λ   into equation [1]:      0    =    1    +    8          (      -      3      )        x                 [1]            x    =          1      24      Substitute -3 for       λ   into equation [2]:      0    =    2    +    2          (      -      3      )        y        y    =          1      3      Use equation [4] to find the value of z:      z    =    4                  (                  1          24                )            2        +                  (                  1          3                )            2            z    =          17      144                  f              (                  1          24                ,                  1          3                ,                  17          144                )              =          1      24        +          2      3        -          51      144                  f              (                  1          24                ,                  1          3                ,                  17          144                )              =          17      48      Note: One cannot use the second derivative to test whether the Lagrange multiplier has given you a maximum or a minimum; the only way to determine whether the value is a local maximum is perturbation of values. I will leave that exercise to you.

How to find the maximum value of the function f(x,y,z)= x+2y-3z subject to the constraint z=4x^2+y^2?

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