If the value of c prescribed bye Lagrange's mean value theorem fo… - Vidyadip

Question

If the value of c prescribed bye Lagrange's mean value theorem for the function
$\text{f}(\text{x})=\sqrt{\text{x}^2-4}$ defined on $[2, 3].$

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Answer

Here,
$\text{f}(\text{x})=\sqrt{\text{x}^2-4}$ defined on $[2, 3].$
We have to find $c$ prescribed by Lagrange's mean value theorem, so
$\text{f}'(\text{c})=\frac{\text{f}(\text{b})-\text{f}(\text{a})}{\text{b}-\text{a}}$
$\Rightarrow\frac{2\text{c}}{2\sqrt{\text{c}^2-4}}=\frac{(\sqrt{9-4})-(\sqrt{4-4})}{3-2}$
$\Rightarrow\frac{\text{c}}{\sqrt{\text{c}^2-4}}=\frac{\sqrt5-0}{1}$
$\Rightarrow\frac{\text{c}}{\sqrt{\text{c}^2-4}}=\sqrt5$
Squaring both sides,
$\Rightarrow c^2 = (c^2- 4)5$
$\Rightarrow 5c^2 - c^2 = 20$
$\Rightarrow 4c^2 = 20$
$\Rightarrow c^2 = 5$
$\Rightarrow\text{c}=\pm\sqrt5$
but $\text{c}=\sqrt5\text{ as }\sqrt5\in(2,3).$

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