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Olympiad workbook

Exam: Olympiad workbook
Subject: IOQM
Chapter: Functions

Olympiad workbook · IOQM · Functions

Let f(x)= (x)/(3)+ (3 x)/(10) for all real x . Find the least...

Let

f(x)=\sin \frac{x}{3}+\cos \frac{3 x}{10}

for all real

x

. Find the least natural number

n

such that

f(n \pi+x)=f(x)

for all real

x

.

Correct answer

60

Step-by-step solution

f(x)=\sin \frac{x}{3}+\cos \frac{3 x}{10}

period of

\sin \frac{x}{3}

is

\frac{6 \pi}{1}

and period of

\cos \frac{3 x}{10}

is

\frac{2 \pi}{3 / 10}=\frac{20 \pi}{3}

LCM is

\frac{60 \pi}{1} \Rightarrow n=60

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