The minimum value of f x = a a x + a 1 - a x , where a , x ∈ R and a > 0 , is equal to:

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Question: Answered & Verified by Expert

The minimum value of

f (x) = a^{a^{x}} + a^{1 - a^{x}}

, where

a, x \in R

and

a > 0

, is equal to:

MathematicsSequences and SeriesJEE MainJEE Main 2021 (25 Feb Shift 2)

Options:

A $a + 1$
B $2 a$
C $a + \frac{1}{a}$
D $2 \sqrt{a}$

Solution:

2195 Upvotes Verified Answer

The correct answer is:

2 \sqrt{a}

$A . M . \geq G . M .$

$f (x) = a^{a^{x}} + a^{1 - a^{x}} = a^{a^{x}} + \frac{a}{a^{a^{x}}} \geq 2 \sqrt{a}$

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