First slide

Functions (XII)

Question

$The period of the function |\sin^{3} \frac{x}{2}| + |\cos^{5} \frac{x}{5}| is$

Moderate

A

$2 π$
B

$10 π$
C

$8 π$
D

$5 π$

Solution

$f (x) = |\sin^{3} \frac{x}{2}| + |\cos^{5} \frac{x}{5}|$
$t h e p e r i o d o f \sin^{3} x i s 2 π (since \sin^{3} (2 π + x) = \sin^{3} x)$
$\Rightarrow t h e p e r i o d o f \sin^{3} \frac{x}{2} = \frac{2 π}{\frac{1}{2}} = 4 π (if f (x) period is p the n f (ax + b) period is \frac{p}{|a|})$
$\Rightarrow s o t h e p e r i o d o f |\sin^{3} \frac{x}{2}| i s 2 π$
$t h e p e r i o d o f \cos^{5} x i s 2 π$
$s o t h e p e r i o d o f \cos^{5} \frac{x}{5} i s \frac{2 π}{\frac{1}{5}} = 10 π$
$s o t h e p e r i o d o f |\cos^{5} \frac{x}{2}| i s = 5 π$
$s o t h e p e r i o d o f f (x) = L C M o f \{2 π, 5 π\} = 10 π$

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