A function f defined on [-2,0] such that f is continuous...

A function $f$ defined on $[-2,0]$ such that $f$ is continuous everywhere except at $x=-2$, differentiable everywhere except at $x=-2$, and fails the conclusion of the Mean Value Theorem with $a=-2$ and $b=0$.

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Key Concepts

Continuity

Continuity is the property of a function that allows small changes in the input to produce small changes in the output. In the context of the Mean Value Theorem (MVT), a function must be continuous on the closed interval for the theorem to apply. Any disruption in continuity, even at an endpoint, can prevent the use of the MVT.

Differentiability

Differentiability means that a function has a defined derivative at each point in an interval, which implies local linearity and smoothness. For the application of the MVT, the function must be differentiable on the open interval, as the existence of a derivative is essential for finding a point where the tangent is parallel to the secant line.

Mean Value Theorem

The Mean Value Theorem (MVT) states that for a function that is continuous on a closed interval and differentiable on the open interval, there exists at least one point in the interval where the instantaneous rate of change (derivative) equals the average rate of change over the interval. The failure to meet the continuity condition, even if differentiability holds elsewhere, invalidates the conclusion of the MVT.

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