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(D) $(i)$ This statement is true,as every continuous function on a closed interval $[a, b]$ is bounded.$(ii)$ This is true by the Intermediate Value T

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only $(i), (ii)$ and $(iv)$ are true

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(D) $(i)$ This statement is true,as every continuous function on a closed interval $[a, b]$ is bounded.$(ii)$ This is true by the Intermediate Value Theorem,as $f(a)  0$.$(iii)$ This is not necessarily true,as the maximum or minimum values can occur at the endpoints $a$ or $b$ where the derivative may not be zero.$(iv)$ This is true. By the Mean Value Theorem,there exists a point $c \in (a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$. Since $f(b) > 0$ and $f(a)  0$,so $f'(c) > 0$.$(v)$ This is not always true. For example,if $f(x) = x$,then $f'(x) = 1 > 0$ for all $x$,so there is no point where $f'(d) Thus,the true statements are $(i), (ii),$ and $(iv)$. The correct option is $(D)$.

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