I'm having trouble understanding how to check the second order conditions for my unconstrained maximization problem.This is the entire problem: Alicia wants to maximize her grade, which is a function of the time spent studying ( T) and the number of cups of coffee ( C) she drinks. Her grade out of 100 is given by the following function. G ( T , C ) = 50 + 10 T + 16 C − ( T 2 + 2 T C + 2 C 2 )In the first order conditions, I find the partial derivatives and set them equal to zero. I get the following two equations: 10 − 2 T − 2 C = 0 and 16 − 2 T − 4 c = 0. The first equation was the partial derivative with respect to T and the second equation was the partial derivative with respect to C. Solving these two equations, I find that C = 3 and T = 2.Now, I need to check the second order conditions. I know that the second partial derivative with respect to both T and C should be negative. This checks out. I get -2 from the first equation (with respect to T) and I get -4 from the second equation (with respect to C). The last thing I need to do with the second order condition is multiply these two together (which yields 8) and then subtract the following: ( δ 2 G δ T δ C ) 2 Please forgive me if this formula isn't displaying correctly. I tried using the laTex equation editor, but I'm not sure if it worked. Anyway, I need to know how to derive this. What is it asking for? I know that this part should be -2 squared, which is 4. Then, 8-4=4, which is positive and tells me that the second order conditions are met.But where is the -2 coming from? I know within both of the equations, there are a few -2's. But, I'm not sure exactly where this -2 comes from.

Question

I&#039;m having trouble understanding how to check the second order conditions for my unconstrained maximization problem.This is the entire problem: Alicia wants to maximize her grade, which is a function of the time spent studying (  T) and the number of cups of coffee (  C) she drinks. Her grade out of 100 is given by the following function.  G  (  T  ,  C  )  =  50  +  10  T  +  16  C  −  (      T    2    +  2  T  C  +  2      C    2    )In the first order conditions, I find the partial derivatives and set them equal to zero. I get the following two equations:  10  −  2  T  −  2  C  =  0 and   16  −  2  T  −  4  c  =  0. The first equation was the partial derivative with respect to   T and the second equation was the partial derivative with respect to   C. Solving these two equations, I find that   C  =  3 and   T  =  2.Now, I need to check the second order conditions. I know that the second partial derivative with respect to both   T and   C should be negative. This checks out. I get -2 from the first equation (with respect to   T) and I get -4 from the second equation (with respect to   C). The last thing I need to do with the second order condition is multiply these two together (which yields 8) and then subtract the following:            (                                    δ            2                    G                          δ          T          δ          C                    )              2      Please forgive me if this formula isn&#039;t displaying correctly. I tried using the laTex equation editor, but I&#039;m not sure if it worked. Anyway, I need to know how to derive this. What is it asking for? I know that this part should be -2 squared, which is 4. Then, 8-4=4, which is positive and tells me that the second order conditions are met.But where is the -2 coming from? I know within both of the equations, there are a few -2&#039;s. But, I&#039;m not sure exactly where this -2 comes from.

Marlee Norman · Accepted Answer

The second-order condition for a maximum of   G  (      x    1    ,  …  ,      x    n    ) says that the Hessian matrix      H          i      j        =                              ∂          2                G                    ∂                  x          i                ∂                  x          j                    is negative semidefinite. So for the case of two variables you need the diagonal elements       H          11        =      ∂    2    G      /    ∂      T    2   and       H          22        =      ∂    2    G      /    ∂      C    2   to be   ≤  0, and the determinant       H          11            H          22        −      H          12        2    ≥  0.

Karina Trujillo · Accepted Answer

Second order equations can always be simplified. In this case the original equation can be rewritten:   84  −  (  C  +  T  −  5      )    2    −  (  C  −  3      )    2  . So, an absolute maximum of 84 at C=3 and T=2.

I'm having trouble understanding how to check the second order conditions for my unconstrained maximization...

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