Question

question
In the mean value theorem f(b) - f(a) =
(b-a)f'(c)if a=4 b=9 f(x)=√x then the value of cis​

Answer 1

$\textbf{Given:}$

$a=4,\;b=9\;\text{and}\;f(x)=\sqrt{x}$

$\textbf{To find:}$

$\text{The value of c using mean value theorem}$

$f(x)=\sqrt{x}$

$f'(x)=\dfrac{1}{2\sqrt{x}}$

$f(a)=f(4)=\sqrt{4}=2$

$f(b)=f(9)=\sqrt{9}=3$

$\textbf{By mean value theorem,}$

$f'(c)=\dfrac{f(b)-f(a)}{b-a}$

$\implies\dfrac{1}{2\sqrt{c}}=\dfrac{3-2}{9-4}$

$\implies\dfrac{1}{2\sqrt{c}}=\dfrac{1}{5}$

$\text{Taking reciprocals, we get}$

$2\sqrt{c}=5$

$\sqrt{c}=\dfrac{5}{2}$

$\,c=\dfrac{25}{4}$

$\implies\,c=6.25\,{\in}\,(4,9)$

$\therefore\textbf{The suitable value of c is 6.25}$

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