Fun with Range

Algebra Level 4

$\large f(x)=2^{x}+2^{-x}+3^{x}+3^{-x}+5^{x}+5^{-x}+3$

What is the range of the function above? Hint: Use the AM-GM Inequality to solve this

[3,\infty)

All real numbers

[9,\infty]

[9,\infty)

[6,\infty )

(6,\infty )

No range

1 solution

Aryan Sanghi
Jul 31, 2020

Let $y = 2^x + 2^{-x} + 3^x + 3^{-x} + 5^x + 5^{-x}$

Using $AM \geq GM$

$\frac{2^x + 2^{-x} + 3^x + 3^{-x} + 5^x + 5^{-x}}{6} \geq \sqrt[^6]{2^x.2^{-x}.3^{x}.3^{-x}.5^x.5^{-x}}$

$2^x + 2^{-x} + 3^x + 3^{-x} + 5^x + 5^{-x} \geq 6\sqrt[^6]{2^x.2^{-x}.3^{x}.3^{-x}.5^x.5^{-x}} = 6$

$\boxed{y \geq 6}$

So, $f(x) = 2^x + 2^{-x} + 3^x + 3^{-x} + 5^x + 5^{-x} + 3 = y + 3$

$y \geq 6$ $y + 3 \geq 9$ $\boxed{f(x) \geq 9}$

Therefore

$\color{#3D99F6}{\boxed{f(x) \in [9, \infty)}}$

Thank you!

Vinayak Srivastava - 10 months, 2 weeks ago