What is minimum total energy? I thought that the energy for a system is constant. But in a question in my physics book we have been asked to determine the minimum total energy from a graph of potential energy vs position? What does this mean?
 A: Technically, the energy of a system is conserved. This is a subtle, but important, distinction from the energy being constant. Conservation is different from constancy in that energy can move in and out of a system and can change forms, but it is neither created nor destroyed.  An expanded formula would look like this:
$$E_{final}=E_{initial}+W_{on}-W_{by}+Q_{in}-Q_{out},$$
where the $E$ values are the total energy of the well-defined system,
$W$ values are work done on the system (increasing or decreasing the system energy) and work done by the system (decreasing the energy),
$Q$ value are energy transfers because of temperature differences with various surroundings.
Constancy would say that $E_{final}=E_{initial}$ always. Conservation says if no force from outside the system does work, the system doesn't work on an outside object, and thermal differences are not considered, $E_{final}=E_{initial}$.
Finally, as @nephente comments, non-relativistically, $E=K+V$ where $V$ explicitly includes some force interaction as part of the system, so $K$ can transform into $V$ and vice-versa without the total energy $E$ changing.
The energy of a rock falling under the influence of a gravitational field can be described in at least two ways:
1) $E = \frac{1}{2} mv^2$, which excludes the action of gravity due to the Earth from the system. In this case, the energy of the rock system is not constant, but the energy is conserved because one must consider the work which the gravity force does on the rock.
2) $E = \frac{1}{2}mv^2 + V_{grav}$. In the simplifying case of negligible air resistance, etc., the energy is both conserved and constant because we don't need to consider the work done by gravity. It is part of the system via the potential energy function.
With a graph such as you described, it is possible for the system to start with different energies, but there will be a minimum because $K$ can't be negative.
