Question

Let $f$ be a differentiable function on the open interval $(a,b)$. Which of the following statements must be true?
$\left(i\right)f$ is continuous on the closed interval $(a,b)$
$\left(ii\right)f$ is bounded on the open interval $(a,b)$
$\left(iii\right)$ If $a<a_1<b_1<b$ and $f\left(a_1\right)<0<f\left(b_1\right)$, then there is a number $c$ such that $a_1<c<b_1$ and $f\left(c\right)=0$

• A. $\left(i\right)$ and $\left(ii\right)$ only
• B. $\left(i\right)$ and $\left(iii\right)$ only
• C. $\left(ii\right)$ and $\left(iii\right)$ only
• D. only $\left(iii\right)$

Answer 1

Answer: (D)

only $\left(iii\right)$

Since the function is differentiable in the open interval $(a,b)$, it will be continuous in the open interval $[a,b]$, but $f\left(a\right)$ may not be equal to $f\left(a^{+}\right)$ or $f\left(b\right)$ may not equal $f\left(b^{-}\right)$
Same is the reason for unboundedness,
However, if applied Intermediate Value Theorem in the open interval $(a,b)$ for $f\left(x\right)$, we get that since $f\left(a_1\right).f\left(b_1\right)<0,f\left(c\right)$ has to be $0$, where $c\in (a_1,b_1)$