Answer
x=4, y=5
The minimum cost for manufacturing a precision camera is $59
Work Step by Step
$L(x,y)=\frac{3}{2}x^{2}+y^{2}-2x-2y-2xy+68$
$L_x(x,y)=3x-2-2y$
$L_y(x,y)=2y-2-2x$
Set each of these equal to 0 and solve for y.
$L_x(x,y)=3x-2-2y=0 \rightarrow y=\frac{3x-2}{2}$
$L_y(x,y)=2y-2-2x=0 \rightarrow y=x+1$
Set them equal, and solve the resulting equation for x.
$\frac{3x-2}{2}=x+1$
$3x-2=2x+2$
$x=4 \rightarrow y=5$
The critical point is (4,5)
$L_{xx}(x,y)=3$
$L_{yy}(x,y)=2$
$L_{xy}(x,y)=-2$
For (4,5)
$L_{xx}(4,5)=3$
$L_{yy}(4,5)=2$
$L_{xy}(4,5)=-2$
$D=3\times2 -(-2)^{2}=2\gt0$ and $L_{xx}(x,y)=3\gt0$, the cost at (4,5) is a minimum
$L(x,y)=\frac{3}{2}x^{2}+y^{2}-2x-2y-2xy+68$
$L(4,5)=59$
The minimum cost for manufacturing a precision camera is $59
