I will use pf/px to be the partial derivative of f wrt x.

In principle f(x,y,z) will have a stationary point (in this case a min because all positive) when pf/px, pf/py, pf/pz are all 0 giving you 3 equations in x,y,z. I tried this but got bogged down with a quartic, so there might be a better approach.

Maybe something like: sqrt(x^2-2xy+y^2+4) reaches a min value of 2 when x=y. etc...

Not sure this helps because sqrt(z^2-24z+340) reaches a min value of 14 when z=12.

There may be a better approach - but I don't see it yet.

x = y does not "cancel" out the first term but, instead, gives it a value of 2. However, if there was a "-4" rather than a "+4" inside the first term then it would indeed give the value of 0. Similarly for y = z.

So we can determine the minimum value of each term but I am not convinced that:

I will use pf/px to be the partial derivative of f wrt x.

In principle f(x,y,z) will have a stationary point (in this case a min because all positive) when pf/px, pf/py, pf/pz are all 0 giving you 3 equations in x,y,z. I tried this but got bogged down with a quartic, so there might be a better approach.

Maybe something like: sqrt(x^2-2xy+y^2+4) reaches a min value of 2 when x=y. etc...

Not sure this helps because sqrt(z^2-24z+340) reaches a min value of 14 when z=12.

There may be a better approach - but I don't see it yet.

Taking x = y = z will cancel out the first 2 terms. The other terms can be minimized by taking z = 12.

This leads to the answer 4*sqrt(10) + 2*sqrt(23).

I was thinking along those same lines. BUT:

x = y = z and z = 12 forces x = 12 and y = 12.

So f(12,12,12)

= 2 + 15 + 4sqrt(10) + 14

=31 + 4sqrt(10)

x = y does not "cancel" out the first term but, instead, gives it a value of 2. However, if there was a "-4" rather than a "+4" inside the first term then it would indeed give the value of 0. Similarly for y = z.

So we can determine the minimum value of each term but I am not convinced that:

sum(mins) = min(sums).

So we are back to

(die)f/(die)x = (die)f/(die)y = (die)f/(die)z = 0