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

Theresa Archer

Theresa Archer

Answered question


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.

Answer & Explanation

Marlee Norman

Marlee Norman

Beginner2022-06-22Added 18 answers

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

Karina Trujillo

Beginner2022-06-23Added 4 answers

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.

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