First slide

Trigonometric Identities

Question

The inequality $\sqrt{x^{\log_{2} \sqrt{x}}} \geq 2$ is satisfied by

Moderate

A

only one value of x
B

$x \in (0, \frac{1}{4}]$
C

$x \in ⌊ 4, \infty)$
D

$x \in (1, 2)$

Solution

$\begin{aligned} \sqrt{x^{\log_{2} \sqrt{x}}} \geq 2, x > 0 \\ \Rightarrow x^{\log_{2} \sqrt{x}} \geq 4 \end{aligned}$
Taking log on both sides with base 2, we get
$\begin{array}{ll} \log_{2} x^{\log_{2} \sqrt{x}} \geq 2 \\ or (\log_{2} \sqrt{x}) (\log_{2} x) \geq 2 \\ or (1 / 2) (\log_{2} x) (\log_{2} x) \geq 2 \\ or {(\log_{2} x)}^{2} \geq 4 \\ or (\log_{2} x - 2) (\log_{2} x + 2) \geq 0 \\ \Rightarrow \log_{2} x \leq - 2 or \log_{2} x \geq 2 \\ \Rightarrow x \leq 1 / 4 or x \geq 4 \\ \Rightarrow x \in (0, 1 / 4] \cup [4, \infty) \end{array}$

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