I have not seen maximum and minimum values in multivariable calculus, but my teacher, but my teacher left as an exercise.
Let
$x+y+z=1$ and $4x^2+9y^2=6$
Find the maximum value of the plane when it intersects with the cylinder.
Could you help me..
methods
Take note that the second equation acts more of like a boundary. So what we could alternatively say is to find the maximum of $f(x,y) = 1-x-y$ on the domain of $4x^2 + 9y^2 = 6$.
Note that we can define the ellipse parametrically as $$x=\sqrt{\frac32}\cos{\theta} $$
$$y= \sqrt{\frac23} \sin{\theta}$$
$$\text{For } 0\le \theta \le 2\pi$$
Now all that is needed to be done is substitute these $x$ and $y$ into $f(x,y)$ to get a new equation $$g(\theta) = 1 - \sqrt{\frac32}\cos{\theta} - \sqrt{\frac23} \sin{\theta}$$
This now becomes a single variable calculus problem, as we now want to find where $g'(\theta) = 0$
This is going to find you your maximum AND minimum, so make sure that once you found your values for $\theta$ you find the corresponding $x$ and $y$ values and plug these into $f(x,y)$ to see which is the maximum and which is the minimum.
I'll let you use this setup to see if you can continue from here :)