Question

The value of $x$ in the interval $[4,9]$ at which the function $f\left(x\right)=\sqrt{x}$ satisfies the mean value theorem is

• A. $\frac{13}{4}$
• B. $\frac{17}{4}$
• C. $\frac{21}{4}$
• D. $\frac{25}{4}$

Answer 1

Answer: (D)

$\left(i\right)f\left(x\right)=\sqrt{x}$ is continuous in $\left[4,9\right]$
$\left(ii\right)f^{\prime }\left(x\right)=\frac{1}{2\sqrt{x}}$
Thus $f\left(x\right)$ is differentiable in $\left(4,9\right)$
$\left(iii\right)f\left(4\right)\ne f\left(9\right)$. All the three conditions of $LMV$ theorem satisfied then there exist at least one $c\in \left(4,9\right)$ such that.
$f^{\prime }\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}\Rightarrow \frac{1}{2\sqrt{c}}=\frac{1}{5}\Rightarrow c=\frac{25}{4}$