I maximized the function f ( x , y ) = ( x + 2 ) ( y + 1 )subject to 4 x + 6 y = 130 , x , y ≥ 0and I found ( x , y ) = ( 16 , 11 ). But the question I'm doing asks me to analyse the "sufficient conditions of second order".So, I thought I should use some theorem that tells me (16,11) is the max value of f. I don't know if I could use Weierstrass' Theorem here and I believe the hessian matrix doesn't help me either.

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I maximized the function  f  (  x  ,  y  )  =  (  x  +  2  )  (  y  +  1  )subject to  4  x  +  6  y  =  130  ,    x  ,  y  ≥  0and I found   (  x  ,  y  )  =  (  16  ,  11  ). But the question I&#039;m doing asks me to analyse the &quot;sufficient conditions of second order&quot;.So, I thought I should use some theorem that tells me (16,11) is the max value of   f. I don&#039;t know if I could use Weierstrass&#039; Theorem here and I believe the hessian matrix doesn&#039;t help me either.

Zain Mccarty · Accepted Answer

Note that the function   f as viewed along the curve   4  x  +  6  y  =  130 only has one degree of freedom (thus only needs one parameter to be expressed). We can write the curve as   y  =            130      −      4      x        6   and use this as a substitution in   f so that  f  (  x  )  =  (  x  +  2  )            (                  130      −      4      x        6    +  1            )      along the curve (yes, specifically along the curve, this is important). We can then look at the second derivative       f    ″    (  x  )  . Notice that only the quadratic term of   f  (  x  ) will contribute to this second derivative so it must be that      f    ″    (  x  )  =            d      2              d              x        2                        [                  −      4              x        2              6              ]        =            −      4                    3        &amp;lt;  0    .This is sufficient to say that   f is concave along the curve   4  x  +  6  y  =  130 and thus an extreme point will achieve a maximum.The distinction here is that we have just computed a sort of &quot;curvature&quot; of   f along   4  x  +  6  y  =  130 whereas the Hessian matrix gives us an idea about the curvature of f as a whole surface.

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