Maximize x 2 − x 1 + y 1 − y 2 given that x 1 2 + y 1 2 = 1 and x 2 2 + y 2 2 = 1.I was thinking about using Lagrange multipliers, but I only know how that works for a 3-variable function, not 4. Could someone please suggest a way to solve this? Maybe with Lagrange multipliers or some more elementary method?

Question

Maximize      x    2    −      x    1    +      y    1    −      y    2  given that       x    1    2    +      y    1    2    =  1 and       x    2    2    +      y    2    2    =  1.I was thinking about using Lagrange multipliers, but I only know how that works for a 3-variable function, not 4. Could someone please suggest a way to solve this? Maybe with Lagrange multipliers or some more elementary method?

Daisy Patrick · Accepted Answer

y    1    −      x    1    ≤            y      1              2              +          x      1              2                  1    +    1    =      2  . Similarly       x    2    −      y    2    ≤      2   so the given expession does not exceed   2      2  . To see that this value is actually attained take       x    1    =  −      1          2      ,       y    1    =      1          2             x    2    =      1          2       and       y    2    =  −      1          2      .

utloverej · Accepted Answer

By the hypotesis you can write       x    1    =  sin  ⁡  θ  ,      y    1    =  cos  ⁡  θ and       x    2    =  sin  ⁡  α  ,      y    2    =  cos  ⁡  α. Then, your want to find the maximum value of  E  =  (  sin  ⁡  α  −  sin  ⁡  θ  )  +  (  cos  ⁡  α  −  cos  ⁡  θ  )  =  (  sin  ⁡  α  +  cos  ⁡  α  )  −  (  sin  ⁡  θ  +  cos  ⁡  θ  )  .But,   −      2    ≤  sin  ⁡  x  +  cos  ⁡  x  ≤      2    ,     ∀  x  ∈  [  0  ,  2  π  ] and the equality holds for   α  =  π      /    4 and   θ  =  5  π      /    4. In particular,   E  ≤  2      2    , exactly as professor Rama Murty found.

Maximize <msub> x 2 </msub> − <msub> x 1 </m

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