<s>[INST] Can't solve lagrange multiplier problem I've been working on this particular problem for some time and can't seem to see what I'm doing wrong. The solution provided says the extrema are $\pm 7^{1/2}/8$. If someone could be so kind as to show me where my error lies, it would be much appreciated.
The questions posed is as follows:


Use lagrange multipliers to find the extrema of $f(x,y,z) = x + y + z$ subject to the constraint $x^2 + 2y^2 + 4z^2 = 1/16$.



I've attached a photo of my work 
Thanks so much!!
 [/INST] So you have
$\lambda2x = \lambda 4y = \lambda 8z = 1\\
y = \frac x2\\
z = \frac x4$
$x^2 + 2 y^2 + 4z^2 = \frac 1{16}\\
x^2 + 2 (\frac x2)^2 + 4(\frac x4) = \frac {1}{16}$
And this is where you made your mistake.  Carrying it through.
$x^2 + \frac {x^2}{2} + \frac {x^2}{4} = \frac {1}{16}\\
16x^2 + 8x^2 + 4x^2 = 1\\
28 x^2 = 1\\
x = \frac {1}{2\sqrt{7}}\\
y = \frac {1}{4\sqrt{7}}\\
z = \frac {1}{8\sqrt{7}}$
$x+y+z = \frac {7}{8\sqrt {7}}\\
x+y+z = \frac {\sqrt 7}{8}$
</s>