First slide

Theory of equations

Question

The real values of x for which $3^{72} {(\frac{1}{3})}^{x} {(\frac{1}{3})}^{\sqrt{x}} > 1$ are

Moderate

A

$x \in [0, 64]$
B

$x \in (0, 64)$
C

$x \in [0, 64)$
D

None of these

Solution

The given inequation is valid only when $x \geq 0 - - - - - (1)$
The given inequation can be written in the form
$\begin{array}{cc} 3^{72 - x - \sqrt{x}} > 1 \\ \Rightarrow & 72 - x - \sqrt{x} > 0 \\ \Rightarrow & x + \sqrt{x} - 72 < 0 \\ \Rightarrow & (\sqrt{x} + 9) (\sqrt{x} - 8) < 0 \\ But & \sqrt{x} + 9 > 0 for all x \geq 0 \\ ∴ & \sqrt{x} - 8 < 0 \Rightarrow \sqrt{x} < 8 \\ ∴ & 0 \leq x < 64 [from (1)] \end{array}$

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