Energy minimisation, confusion in interpretation of question

[Dazed&Confused]

Perhaps not an appropriate place to ask this. I've completed a question which is phrased as the following:

Explain, on thermodynamical grounds, why the minimisation of the energy $E(S, V, N)$ yields the equilibrium state of a system with fixed entropy S, volume V and number of particles $N$.

I had one interpretation, to show that minimisation of the energy, i.e. that $dE = 0$, leads to a fixed entropy, volume, and number of particles.

The correct interpretation was actually to show that under fixed, entropy, and number of particles the energy is minimised, using the second law. To me, this does not make sense, however it did lead to one question: is there a situation where minimising $E$ would lead to a constant entropy, volume, and number of particles where this was not specified initially?

 

[Simon Bridge]

The correct interpretation was actually to show that under fixed, entropy, and number of particles the energy is minimised, using the second law.

... that's how I read it. The state is given by the numbers (S,V,N).

To me, this does not make sense, however it did lead to one question: is there a situation where minimising $E$ would lead to a constant entropy, volume, and number of particles where this was not specified initially?

... what would the constraint be? Minimum energy could be zero for zero particles.

 

[Dazed&Confused]

Ok thanks. Ignore my question.

What I do not understand is why if $E(S,V,N)$ does constant $S, V, N$ not mean constant $E$.

I've seen the derivation using the availability but I'm not sure why the supposed contradiction isn't one.