The domain of definition of the function $f(x)=\sqrt[6]{4^x+8^{\frac{2}{3}(x-2)}-52-2^{2(x-1)}}$ is:
Answer & explanation
Correct answer: option 3
$4^x+8^{\frac{2}{3}(x-2)}-52-2^{2(x-1)}>0$
$⇒2^{2x}+2^{2(x-2)}-52-2^{2(x-1)}>0$
$⇒2^{2x}\left[1+\frac{1}{16}-\frac{1}{4}\right]>52⇒2^{2x}\left[\frac{16+1-4}{16}\right]>52$
so $2^{2x}\left[\frac{13}{16}\right]>52$
$2^{2x}>64$
so $x ≥3 ⇒x∈[3, \infty)$
Hence (3) is the correct answer.