Which of the functions f satisfies neither the hypotheses...

Which of the functions $f$ satisfies neither the hypotheses nor the conclusions of LMVT
(A) $f(x)=|x-2| ;[1,4]$
(B) $f(x)=1+|x-1| ;[0,3]$
(C) $\mathrm{f}(\mathrm{x})=[\mathrm{x}]$ (the greatest integer function); $[-1,1]$
(D) $f(x)=3 x^{2 / 3} ;[-1,1]$

Key Concepts

Verification of LMVT Hypotheses

Before concluding any result from the LMVT, it is crucial to check that a function meets its hypotheses—namely, continuity on the closed interval and differentiability on the open interval. If either condition fails, then the theorem cannot be applied, and the anticipated conclusion (the existence of a point where the derivative equals the average slope) may not hold.

Differentiability

Differentiability refers to the existence of a derivative at every point in an open interval. It indicates that the function has a well-defined tangent at every point in that interval. For LMVT to apply, the function must be differentiable on the open interval; any points where the function is not differentiable (such as sharp corners or cusps) mean that the hypothesis of the theorem is not completely met.

Lagrange Mean Value Theorem

The Lagrange Mean Value Theorem (LMVT) is a fundamental result in calculus that states if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point in the interior where the function’s instantaneous rate of change (its derivative) equals the average rate of change over the entire interval.

Continuity

Continuity is a property of functions whereby small changes in the input result in small changes in the output, meaning the function has no breaks, jumps, or points of discontinuity on the interval in question. This property is a necessary hypothesis for applying the LMVT, as the theorem only holds when the function is continuous on the entire closed interval.

Transcript

Please give Ace some feedback