Find the range of the function $f(x)$ if $f(x) = 2^x + \frac{4}{2^x}$

Question

I tried this by a logical approach as the sum of two positive numbers is constant will be minimum if they are equal , i.e. $\frac{4}{2^x}$ each should be equal to $2.$ Hence minimum value will be $4.$ The range is $[4,\infty)$ What is proper way of solving these kinds of problem ?

Answer 1

By using the concept of AM > or equal to GM Since both functions $2^x$ and $4^x$ are positive and their product is finite value ($=4$) we can apply AM>GM .

(2^x + 4/2^x)/2 > or equal to (2^x*4/2^x)^1/2 => 2^x + 4/2^x > or equal to $4 . $

Hence the range is $[4,\infty).$