Step 1: Recall the corner-point rule.
In linear programming the objective function is linear, and a linear function over a polygon-shaped feasible region always reaches its best value at a corner (vertex) of that region. This is the heart of the corner-point method.
Step 2: Test statement (A).
An optimal solution does not always exist. If the feasible region is empty, or unbounded in the wrong direction, there may be no optimum. So (A) is false.
Step 3: Test statement (B).
If the region is unbounded, a maximum or minimum may fail to exist because the objective can keep growing or falling without limit. So (B) is not always true.
Step 4: Test statement (C).
A bounded region does guarantee both a maximum and a minimum, but the way (C) is worded suggests only one of them, so it is not the cleanest correct statement here.
Step 5: Test statement (D).
Whenever an optimal value does exist, it must occur at a corner point of the feasible region. This is exactly the fundamental theorem of LPP and is always true.
Step 6: Pick the correct option.
The only statement that is always correct is (D). \[ \boxed{\text{Optimal value, if it exists, occurs at a corner point}} \]