Ques : maximize $$F=xy+\pi x^2/8+k(2y+x+\pi x/2 - 40)$$ I tried solving it using Lagrange multiplier method to get the answer $$(x,y)=(80/(\pi+4),40/(\pi+4))$$ which is correct as per book solution. I was trying to verify whether it corresponds to maximum or minimum value. So,
$$F_{xx}=\pi /4$$ $$F_{yy}=0$$ $$F_{xy}=1$$
Now $$F_{xx}F_{yy}-F_{xy}^2 < 0$$ This means $(x,y)=(80/(\pi+4),40/(\pi+4))$ is neither maximum nor minimum. Where am I going wrong?
Along the line
$$\left(1+\frac{\pi}{2}\right)x+2y=40$$
the function
$$ f(x,y)=xy+\frac{\pi}{8}x^2 $$
reduces to
$$ f(x)=20x-\left(\frac{1}{2}+\frac{\pi}{8}\right)x^2 $$
which attains a maximum value at $x=\dfrac{80}{4+\pi}$
The second derivative test would be appropriate if one were determining the nature of the critical points of $f(x,y)$. [which is at $(0,0)$.]