First slide

Introduction to real valued functions

Question

Number of solutions of equation $4^{x} - 3^{x - \frac{1}{2}} = 3^{x + \frac{1}{2}} - 2^{2 x - 1} is$

Easy

A

0
B

1
C

2
D

infinite

Solution

$\begin{array}{l} 4^{x} - 3^{x - \frac{1}{2}} = 3^{x + \frac{1}{2}} - 2^{2 x - 1} \\ \Rightarrow 4^{x} - \frac{3^{x}}{\sqrt{3}} = 3^{x} \sqrt{3} - \frac{4^{x}}{2} \\ \Rightarrow \frac{3}{2} \cdot 4^{x} = 3^{x} (\sqrt{3} + \frac{1}{\sqrt{3}}) \\ \Rightarrow \frac{3}{2} \cdot 4^{x} = 3^{x} \cdot \frac{4}{\sqrt{3}} \\ \Rightarrow \frac{4^{x - 1}}{4^{1 / 2}} = \frac{3^{x - 1}}{\sqrt{3}} \\ \Rightarrow 4^{x - 3 / 2} = 3^{x - 3 / 2} \\ \Rightarrow {(\frac{4}{3})}^{x - 3 / 2} = 1 \\ \Rightarrow x - \frac{3}{2} = 0 \\ \Rightarrow x = 3 / 2 \end{array}$

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