The minimum value of

Question:

The minimum value of $f(x)=a^{a^{x}}+a^{1-a^{x}}$ where $\mathrm{a}, \mathrm{x} \in \mathrm{R}$ and $\mathrm{a}>0$, is equal to :

  1. $2 \mathrm{a}$

  2.  $2 \sqrt{\mathrm{a}}$

  3. $a+\frac{1}{a}$

  4. $a+1$


Correct Option: , 2

Solution:

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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