The minimum positive value of the expression $(10 - 5^{- x} - 5^{x})^{- 1}$ is

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Solution

$(10 - 5^{- x} - 5^{x})^{- 1} = \frac{1}{(10 - 5^{- x} - 5^{x})} = \frac{1}{10 - \frac{1}{5^{x}} - 5^{x}} = \frac{1}{10 - (\frac{1}{5^{x}} + 5^{x})}$
$\frac{1}{5^{x}} + 5^{x} \geq 2$
$10 - (\frac{1}{5^{x}} + 5^{x}) \leq 8$
$10 - (\frac{1}{5^{x}} + 5^{x})$ max value is $8$
So minimum positive value of $(10 - 5^{- x} - 5^{x})^{- 1}$ is $0.125$

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