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The minimum value of

Question:

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

  1. (1) $\mathrm{a}+\frac{1}{\mathrm{a}}$

  2. (2) $a+1$

  3. (3) $2 \mathrm{a}$

  4. (4) $2 \sqrt{a}$

Correct Option: , 4

Solution:

$\mathrm{AM} \geq \mathrm{GM}$

$\frac{\mathrm{a}^{\mathrm{ax}}+\frac{\mathrm{a}}{\mathrm{anx}}}{2} \geq\left(\mathrm{a}^{\mathrm{ax} \cdot} \frac{\mathrm{a}}{\mathrm{a}^{\mathrm{ax}}}\right)^{1 / 2} \Rightarrow \mathrm{a}^{\mathrm{ax}}+\mathrm{a}^{1-\mathrm{ax}} \geq 2 \sqrt{\mathrm{a}}$

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