- #1

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- Homework Statement:
- Find the range of ##f(x) = 6^x + 3^x + 6^{-x} + 3^{-x} + 2##

- Relevant Equations:
- AM >= GM

##f(x) = 6^x + 3^x + 6^{-x} + 3^{-x} + 2##

But, ## AM >= GM##

So,

##f(x) >= 5 * 2 ^ {\frac{1}{5}}##

But this is not the case. According to the graph, it is ## f(x) >= 6##.

If I do the same thing without considering the constant '2' then I am getting the answer.

let ##g(x) = 6^x + 3^x + 6^{-x} + 3^{-x} ## and ##f(x) = g(x) + 2##

Using ##AM>=GM##,

##g(x) >= 4##

Hence, ##f(x) >= 6##

Why is this the case? Why is the latter approach working but no the former?

But, ## AM >= GM##

So,

##f(x) >= 5 * 2 ^ {\frac{1}{5}}##

But this is not the case. According to the graph, it is ## f(x) >= 6##.

If I do the same thing without considering the constant '2' then I am getting the answer.

let ##g(x) = 6^x + 3^x + 6^{-x} + 3^{-x} ## and ##f(x) = g(x) + 2##

Using ##AM>=GM##,

##g(x) >= 4##

Hence, ##f(x) >= 6##

Why is this the case? Why is the latter approach working but no the former?