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$f (x) = \log [\frac{x^{2} + a b}{(a + b) x}]$ on $[a, b]$ such that $a > 0$ and $b > 0$ then value of c in RMVT is

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$- \sqrt{a / b}$

$\sqrt{a / b}$

$- \sqrt{a b}$

$\sqrt{a b}$

answer is D.

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Detailed Solution

$f (x) = \log (\frac{x^{2} + a b}{(a + b) x})$
$f^{1} (x) = \frac{1}{x^{2} + a b} (a + b) x . (a + b) x (2 x) - (x^{2} + a b) (a + b) = 0$
$2 x^{2} - x^{2} - a b = 0$
$x^{2} = a b ∴ c = \sqrt{a b}$

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