Verify that f satisfies the conditions of the mean-value...

Verify that $f$ satisfies the conditions of the mean-value theorem on the indicated interval and find all numbers $c$ that satisfy line conclusion of the theorem.
$$f(x)=\sqrt{1-x^{2}} ; \quad[0.1]$$

Key Concepts

Mean Value Theorem

The Mean Value Theorem is a fundamental result in calculus which states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one number c in (a, b) where the instantaneous rate of change (the derivative at c) equals the average rate of change over [a, b]. This theorem connects the behavior of a function’s derivative with its overall change on an interval.

Continuity on Closed Intervals

Continuity on a closed interval [a, b] means that the function has no breaks, jumps, or removable discontinuities in that interval and attains every value between f(a) and f(b). Establishing continuity on [a, b] is essential for applying the Mean Value Theorem because it guarantees there is a smooth transition from one endpoint to the other.

Differentiability on Open Intervals

Differentiability on an open interval (a, b) requires that at every point inside the interval, the function has a defined derivative, ensuring that it behaves smoothly without any sharp corners or cusps. This condition is crucial for the Mean Value Theorem since the theorem’s guarantee is on the existence of a point where the instantaneous rate of change matches the average rate of change.

Average Rate of Change

The average rate of change of a function over an interval [a, b] is computed as (f(b) - f(a)) divided by (b - a). This value represents the overall change in the function per unit change in x over that interval and serves as a benchmark for comparing to the instantaneous rate of change at a specific point guaranteed by the Mean Value Theorem.

Solving for the Point c

Once the conditions of the Mean Value Theorem are verified and the average rate of change is determined, the next step is to set the derivative f'(c) equal to this average rate of change and solve for c. This process involves using differentiation techniques and algebraic manipulation to isolate c, thereby identifying the specific point(s) in the interval where the function’s instantaneous rate of change equals the overall rate of change.

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