The maximum value of $3 - e^{3 x} - e^{- 3 x}$ is

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Solution

$3 - e^{3 x} - e^{- 3 x} = 3 - (e^{3 x} + \frac{1}{e^{3 x}})$
Here, $e^{3 x} + \frac{1}{e^{3 x}}$ is in the form of $x + \frac{1}{x}$ where $x > 0$
So, minimum value of is $e^{3 x} + \frac{1}{e^{3 x}}$ is $2.$
$∴$ Maximum $3 - e^{3 x} - e^{- 3 x}$ is $1.$

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