I am not quite sure what you want here. If you simply want a 3-D plot of f(x,y) over the domain where 9x^2+16y^2≤144, then here is one way.
1) Let's plot the region of the Domain first:
So now we see what it looks like.
2) Define a function that returns the value of f(x,y) when 9x^2+16y^2≤144 and zero otherwise. This way we can plot the surface to see what it look like. We want numerical values so let's make sure the fcn returns a number and not an equation. So:
3) Plot said fcn:
I have added a slider so I can measure the z value of the min and max.
4) Having said that we can solve for the min and max numerically. We see that the min & max are naturally enough along the boundary where 9x^2+16y^2 = 144. So let's make a function that traces this boundary. Easy way is to solve for y as a function of just x So:
Since the solution is of a quadratic, we'll keep both the solutions. We know from our 3-D plot that the max value is in the fourth quadrant and the min is in the first.
We can check our work by overlaying our implicit plot with our eqs
Now we can solve for the min and max. Easiest way is to take the derivative of the two equations and solve for zero, so:
Hopefully that is useful.