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(D) Statement $I$ is false because differentiability on an open interval does not guarantee continuity at the endpoints.Statement $II$ is false. Consi

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only $III$

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(D) Statement $I$ is false because differentiability on an open interval does not guarantee continuity at the endpoints.Statement $II$ is false. Consider the function $f(x) = 1/x$ on the interval $(0, 1)$. This function is differentiable on $(0, 1)$ but is not bounded as $x 	o 0^+$.Statement $III$ is true. Since $f$ is differentiable on $(a, b)$,it is continuous on $(a, b)$. Therefore,$f$ is continuous on the closed sub-interval $[a_1, b_1]$. By the Intermediate Value Theorem,since $f(a_1) < 0 < f(b_1)$,there must exist at least one $c \in (a_1, b_1)$ such that $f(c) = 0$.

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