The least value of $e^{s i n^{- 1} x} + e^{c o s^{- 1} x}$ is

1 + e^{π / 2}

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2 e^{π / 2}

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e^{π / 4}

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2 e^{π / 4}

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Solution

The correct option is D $2 e^{π / 4}$
For positive numbers we have,
$A M \geq G M$
$\frac{e^{s i n^{- 1} x} + e^{c o s^{- 1} x}}{2} \geq \sqrt{e^{(s i n^{- 1} x + c o s^{- 1} x)}}$
$e^{s i n^{- 1} x} + e^{c o s^{- 1} x} \geq 2 e^{π / 4}$ $[s i n^{- 1} x + c o s^{- 1} x = \frac{π}{2}]$
Hence, option 'D' is correct.

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