MCQ
Let $f$ be $a$ differentiable function on the open interval $(a, b)$. Which of the following statements must be true?

$I$. $f$ is continuous on the closed interval $[a, b]$

$II.$ $f$ is bounded on the open interval $(a, b)$

$III.$ If $a$ $< a_1< b_1< b$, and $f (a_1)<0< f (b_1)$, then there is $a$ number $c$ such that $a_1 < c < b_1$ and $f (c)=0$

  • A
    $I$ and $II$ only
  • B
    $I$ and $III$ only
  • C
    $II$ and $III$ only
  • only $III$

Answer

Correct option: D.
only $III$
d
$I$ and $II$ are false.

The function $f (x) = 1/x, 0 < x < 1$, is a counter example.

Statement $III$ is true. Apply the intermediate value theorem to $f$ on the closed interval $[a_1, b_1]$

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