Question

The range of the function:-
$f(x) = {2}^{x } + {4}^{x} + {2}^{ - x} + {4}^{ - x}$
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Answer 1

$\large\underline{\text{Solution}}$

$\red{\bigstar}$ A.M-G.M inequality

If the given A.M or G.M is a constant, the maximum or minimum can be found. However, this requires the numbers to be positive or zero.

In the question, this concept will be used as the G.M of each pair of numbers is a constant and numbers are positive.

By A.M-G.M inequality

$\implies2^{x}+2^{-x}\geq2\sqrt{2^{x}\cdot2^{-x}}$

$\implies2^{x}+2^{-x}\geq2$

(equals when $x=0$)

In the same way

$\implies4^{x}+4^{-x}\geq2\sqrt{4^{x}\cdot4^{-x}}$

$\implies4^{x}+4^{-x}\geq2$

(equals when $x=0$)

Hence

$\implies2^{x}+2^{-x}+4^{x}+4^{-x}\geq4$

that is

$\implies f(x)\geq4$

which minimum occurs at $x=0$.

$\large\underline{\text{Conclusion}}$

The range of the function is $f(x)\in[4,\infty)$.

Answer 2

Step-by-step explanation:

given :

• The range of the function:-
• $f(x) = {2}^{x } + {4}^{x} + {2}^{ - x} + {4}^{ - x}$

to find :

• The range of the function

solution :

• the answer in attachment please check it