Final answer:
To find the minimum or maximum values within a bounded region like the one bounded by a parabola, the Extreme Value Theorem is most useful. This theorem assures that a continuous function on a closed interval will have both extreme values, and they can be found by examining the function's derivative and behavior at the boundaries of the region.
Step-by-step explanation:
If you seek the minimum or maximum value of a function within a bounded region, such as the region bounded by the parabola y=3-x², the Extreme Value Theorem is of great utility. This theorem tells us that if a function is continuous on a closed interval, then it must have both a minimum and a maximum value on that interval. Thus, within the context of a closed and bounded region, you can be sure that the function will have both extreme values, and these can be found by examining where the function's gradient (represented by the derivative) is zero (indicative of critical points) and by assessing the function's behavior at the boundaries of the region.
While there are other techniques such as the Gradient Descent method, which is typically used for unconstrained optimization and therefore is less suited for bounded regions, or the Method of Lagrange Multipliers, which is more suitable for constrained optimization problems typically involving equality constraints, the Extreme Value Theorem directly addresses finding extreme values on bounded regions. The Chain Rule is relevant in differentiating composite functions and is not specifically targeted towards finding extrema within bounded regions.