I feel like im drowning this site with question about implicit function theorem but I really do not understand how I can find the differential.
we are given elipsoid $x^2+y^2+z^2+xy+yz-54=0$
We are asked to show that the maximal value of $z$ on the ellipsoid is at $(x,y,z)=(3,-6,9)$ using the implicit function theorem
if we define $F(x,y,z)=x^2+y^2+z^2+xy+yz-54$ we will see that the gradient (differential) is:
$(2x+y,2y+x+z,2z+y)$ meaning if $z \neq -\frac{y}{2}$ then we can represent $z=z(x,y)$ as a function of $x$ and $y$ according to implicit function theorem.
the maximal value of $z$ will be attained when $z'(x,y)=0$ or at a point where $z=-\frac{y}{2}$ (at which case, we cant represent $z$ as a function of $x$ and $y$.)
how do we find the differential of $z$?
I mean we want, $\frac{\partial z}{\partial x} = -\frac{F_x}{F_z} = \frac{-2x-y}{2z+y}=0$ and $\frac{\partial z}{\partial y} = -\frac{F_y}{F_z} = \frac{-2y-x}{2z+y}=0$
That would imply $x=0,y=0$ and $z=\sqrt{54}$. But perhaps there are higher values of $z$ where we cant express $z$ as a function...how do we find those?
I miscalculated.
$(x,y,z)=(3,-6,9)$ is the correct answer. the $\frac{\partial z}{\partial y}$ written in the question is wrong.