optimization finding value to maximize

Question

I have assignment about optimization.

Q. the profit per acre of a farm is given by $f(x_1,x_2)= 20x_1+ 26x_2 + 4x_1x_2-4x_1^2-3x_2^2$ where $x_1$ and $x_2$ denote, respectively, the labor cost and the fertilizer cost. Find value of $x_1$ and $x_2$ that maximize the profit.

I use $grad f=0$ and hessian matrix. but I can't find eigenvalue of hessian matrix..

What;s wrong with my approah?? I need master to help me ...

Answer 1

Now $\frac{\partial f}{\partial x_1}=20+4x_2-8x_1$ and $\frac{\partial f}{\partial x_2}=26+4x_1-6x_2$.

We can solve $\frac{\partial f}{\partial x_1}=\frac{\partial f}{\partial x_2}=0$ and get $(x_1,x_2)=(7,9)$

Now we check that it's indeed a maximum: $\frac{\partial^2 f}{\partial x_1^2}=-8$, $\frac{\partial^2 f}{\partial x_2^2}=-6$ and $\frac{\partial^2 f}{\partial x_1 x_2}=4$ so the Hessian matrix has determinant $D=(-8)(-6)-4^2=48-16=32 > 0$ and $\frac{\partial^2 f}{\partial x_1^2} < 0$ and we can conclude that $(7,9)$ is a maximum.

Answer 2

Let $x_1=x$ and $x_2=y$.

Thus, we need to find a maximal value of $$20x+26y+4xy-4x^2-3y^2.$$ Indeed, $$20x+26y+4xy-4x^2-2y^2=187-(2x-y-5)^2-2(y-9)^2\leq187.$$ The equality occurs for $y=9$ and $2x-y-5=0,$ which says that we got a maximal value.

Id est, we got the answer: $(7,9).$