1) You really should have simplified your conditions first by dividing out 2 and 4 before differentiating it. This does not change the result, but it is needless complication.
2) $x$, $y$ and $z$ are positive numbers, so we have to take into account the behaviour on the border when one or more of them are zero.
The situation is symmetric, so say $z=0$, but then you have $x+y=50$ and $xy= 750$ which gives (by the solution formula of the quadratic equation) that $x$ and $y$ are not real numbers.
This means that subject to the two conditions there is actually no border and every extrema can be found by using Lagrange multipliers.
3)
You write:
$x(y−z)/2=λ(y−z)$, so that $λ=x/2$
But what you get, for each of the three similar equations, is:
Either (a) $y=z$ or (b) $\lambda=x/2$.
Either (a) $z=x$ or (b) $\lambda=y/2$.
Either (a) $x=y$ or (b) $\lambda=z/2$.
Now, you would have to choose (a) or (b) for each of the three equations which gives 8 cases (aaa,aab,aba,abb,baa,bab,bba,bbb), but it is simpler than that.
If you choose at least once condition (a), then you have two equal variables. If you choose at least twice condition (b), then you have two variables equal to $2\lambda$, so you also have two equal variables.
So, whatever you do, two variables will be equal.
But the problem statement is symmetric in $x$, $y$ and $z$ (which means that if you interchange the three variables, the problem statement does not change, for example $yzx=xyz$, so the volume you are looking for will not depend on the order of the variables).
Therefore, since you know that two variables are equal and you know that it doesn't really matter for the end result which ones are equal, it suffices to investigate $x=y$:
(At this point, we still might have $x=y=z$, but we don't know that. What we do know, is that the original conditions are true, so we use them first, and it turns out that they already fix all the values.)
The original conditions give:
$2x+z=50$ and $x^2+2xz=750$.
Substituting $z=50-2x$ in the second equation gives a quadratic equation $3x^2-100x+750=0$.
This gives $x=\frac{50\pm 5\sqrt{10}}{3}$.
You find two solutions (up to symmetry):
$x=y=\frac53 (10 + \sqrt{10})$, $z=\frac{10}3(5-\sqrt{10})$ giving the volume
$\frac{2500}{27}(35-\sqrt{10})\approx 2948$
and
$x=y=\frac 53 (10- \sqrt{10})$, $z=\frac{10}3(5+\sqrt{10})$ giving the volume
$\frac{2500}{27}(35+\sqrt{10})\approx 3534$.
Since these are the only two candidates for extrema, the first one is the minimum value and the second one is the maximum value.
Best Answer
The volume of a parallelepiped is indeed $V=xyz$. If it is inscribed into an ellipse $x^2/a^2+y^2/b^2+z^2/c^2=1$, then $z$ in the volume formula can be replaced by $z$ from the ellipse. Now the picture only shows $1/8$th part (there are $8$ octants) of the whole parallelepiped. Hence: $V_{large}=8V_{small}$. So there are two ways to find the volume of the large parallelepiped: direct or indirect: $$Direct: \ \ V_{large}=2x\cdot 2y\cdot 2\cdot c\sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}.$$ $$Indirect: \ \ V_{large}=8V_{small}=8\cdot x\cdot y\cdot c\sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}.$$ Note: It looks one of two $c$s is excessive in the volume formula you provided. However, $f_x, f_y$ were found with one $c$.
Appendix: $$f_x=yc\sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}+xyc\cdot \frac{-\frac{2x}{a^2}}{2\sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}}=yz-\frac{x^2yc^2}{a^2z}=\frac{y(a^2z^2-c^2x^2)}{a^2z}.$$ Note: After finding $f_x$ the root was replaced by $z$.