First slide

Types of real valued functions XI

Question

If $\log_{2} x + \log_{2} y \geq 6$ then the least value of x + y is

Moderate

A

4
B

8
C

16
D

32

Solution

$Given \log_{2} x + \log_{2} y \geq 6$
$\begin{array}{ll} \Rightarrow \log_{2} (xy) \geq 6 \\ \Rightarrow xy \geq 64 \end{array}$
Also to define $\log_{2} x and \log_{2} y$ we have
x>0,y>0.
Since A.M. $\geq$ G.M., we have
$\begin{array}{l} \frac{x + y}{2} \geq \sqrt{xy} \\ or x + y \geq 2 \sqrt{xy} \geq 16 \end{array}$

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