Question

In the mean value theorem $f(b)-f(a)=(b-a)f^{\prime }(c),$ if $a=4,b=9$ and $f(x)=\sqrt{x},$ then the value of c is

• A. $8.00$
• B. $5.25$
• C. $4.00$
• D. $6.25$

Answer 1

Answer: (D)

Given, $a=4,n=9,f(x)=\sqrt{x}$ and $f(b)-f(a)=(b-a)f^{\prime }(c)$
$\Rightarrow$ $f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}=\frac{3-2}{9-4}$
$\Rightarrow$ $f^{\prime }(c)=\frac{1}{5}$
$\Rightarrow$ $\frac{1}{2}{c}^{-1/2}=\frac{1}{5}$
$\Rightarrow$ $c^{-1/2}=\frac{5}{2}$
$\Rightarrow$ $c={\left(\frac{5}{2}\right)}^2=\frac{25}{4}=6.25$