Theresa Archer

2022-06-21

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.
$G\left(T,C\right)=50+10T+16C-\left({T}^{2}+2TC+2{C}^{2}\right)$
In the first order conditions, I find the partial derivatives and set them equal to zero. I get the following two equations:

$10-2T-2C=0$ and $16-2T-4c=0$. 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:
${\left(\frac{{\delta }^{2}G}{\delta T\delta C}\right)}^{2}$
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.

Marlee Norman

The second-order condition for a maximum of $G\left({x}_{1},\dots ,{x}_{n}\right)$ says that the Hessian matrix
${H}_{ij}=\frac{{\mathrm{\partial }}^{2}G}{\mathrm{\partial }{x}_{i}\mathrm{\partial }{x}_{j}}$
is negative semidefinite. So for the case of two variables you need the diagonal elements ${H}_{11}={\mathrm{\partial }}^{2}G/\mathrm{\partial }{T}^{2}$ and ${H}_{22}={\mathrm{\partial }}^{2}G/\mathrm{\partial }{C}^{2}$ to be $\le 0$, and the determinant ${H}_{11}{H}_{22}-{H}_{12}^{2}\ge 0$.

Karina Trujillo

Second order equations can always be simplified. In this case the original equation can be rewritten: $84-\left(C+T-5{\right)}^{2}-\left(C-3{\right)}^{2}$. So, an absolute maximum of 84 at C=3 and T=2.