Final answer:
The Mean Value Theorem will hold for a function y(x) if it is continuous on a closed interval [a, b] and differentiable on the open interval (a, b). It guarantees that there is at least one point in (a, b) where the instantaneous rate of change equals the average rate of change over that interval.
Step-by-step explanation:
The Mean Value Theorem (MVT) in calculus is a fundamental theorem that connects the average rate of change of a function over an interval with the instantaneous rate of change at a point within that interval. For the theorem to hold, there are two important conditions that must be satisfied:
- The function y(x) must be continuous on the closed interval [a, b].
- The function must be differentiable on the open interval (a, b), which means the derivative dy(x)/dx must exist at every point in that interval.
If these conditions are met, the MVT states there exists at least one point c in the open interval (a, b) where the instantaneous rate of change (the derivative dy(c)/dx) is equal to the average rate of change ((y(b) - y(a))/(b - a)) over [a, b].
However, if the derivative does not exist at every point on (a, b), or if y(x) is not continuous on [a, b], then the Mean Value Theorem does not necessarily apply.