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(A) Lagrange's Mean Value Theorem $(LMVT)$ is applicable to a function $f(x)$ on $[a, b]$ if: $1$. $f(x)$ is continuous on $[a, b]$. $2$. $f(x)$ is di

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$III, IV$

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(A) Lagrange's Mean Value Theorem $(LMVT)$ is applicable to a function $f(x)$ on $[a, b]$ if: $1$. $f(x)$ is continuous on $[a, b]$. $2$. $f(x)$ is differentiable on $(a, b)$. Let us analyze each function on $[0, 1]$: $I) f(x) = \begin{cases} \frac{1}{2}-x, & x  \frac{1}{2}$. At $x = \frac{1}{2}$,$LHD = -1$ and $RHD = 0$. Since $LHD 
eq RHD$,it is not differentiable at $x = \frac{1}{2}$. $II) f(x) = |3x-1|$. This function is not differentiable at $x = \frac{1}{3} \in (0, 1)$. $III) f(x) = x|x|$. This is $x^2$ for $x \geq 0$ and $-x^2$ for $x $IV) f(x) = |x|$. This function is not differentiable at $x = 0$. Since $0$ is an endpoint,we check the interval $(0, 1)$. It is differentiable on $(0, 1)$. However,standard $LMVT$ requires differentiability on the open interval $(a, b)$. While $f(x) = |x|$ is differentiable on $(0, 1)$,it is often excluded in strict contexts if the derivative does not exist at the boundary. Given the options,$III$ is definitely correct. Comparing with options,$I$ and $III$ is the most suitable choice as $f(x)$ in $I$ is continuous on $[0, 1]$ and differentiable on $(0, 1) \setminus \{\frac{1}{2}\}$. Actually,$III$ is the only one strictly differentiable on $(0, 1)$. Re-evaluating: $I$ is not differentiable at $1/2$. $II$ is not at $1/3$. $IV$ is differentiable on $(0, 1)$. Thus,$III$ and $IV$ are the best candidates.

Consider the following functions:
$I) f(x) = \begin{cases} \frac{1}{2}-x, & x < \frac{1}{2} \\ (\frac{1}{2}-x)^2, & x \geq \frac{1}{2} \end{cases}$
$II) f(x) = |3x-1|$
$III) f(x) = x|x|$
$IV) f(x) = |x|$
Then on $[0, 1]$,Lagrange's Mean Value Theorem $(LMVT)$ is applicable to which of the functions?

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Consider the following functions:$I) f(x) = \begin{cases} \frac{1}{2}-x, & x < \frac{1}{2} \\ (\frac{1}{2}-x)^2, & x \geq \frac{1}{2} \end{cases}$$II) f(x) = |3x-1|$$III) f(x) = x|x|$$IV) f(x) = |x|$Then on $[0, 1]$,Lagrange's Mean Value Theorem $(LMVT)$ is applicable to which of the functions?