Q. If the range of the function $f\left(x\right)=6^{x}+3^{x}+6^{- x}+3^{- x}+2$ is $\left[k , \infty $ , then the value of $k$ is

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Solution:

We have, $f\left(x\right)=6^{x}+3^{x}+6^{- x}+3^{- x}+2$
Since, $6^{x}+6^{- x}\geq 2$ and $3^{x}+3^{- x}\geq 2$
Therefore, $f\left(x\right)\geq 2+2+2$
$\Rightarrow f\left(x\right)\geq 6$
Thus, $f\left(x\right)\in \left[6 , \infty \right)$
Hence, the value of $k$ is $6.$

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