Question

Prove that f(x)=x^2 is not periodic

Answer 1

ANSWER

Period of ∣sin3x∣ is

3

π

and periode of sin

2

x is π

(1) Same as the periode of ∣sin3x∣ or

2

1−cos2x

whose period is π

Now period of ∣sin3x∣+sin

2

x is the L.C.M of their periods

∴ L.C.M of (

3

π

,π)=

H.C.F(3,1)

L.C.M(π,π)

=π

(2) Similarly we can say that cos4x+tan

2

x and cos2x+sinx are periodic function.

(3)Now cos

2

x is periodic with periode π and for periode of cos

x

Let f(x)=cos

x

Let f(x+T)=f(x)

⇒cos

T+x

=cos

x

⇒

T+x

=2nπ±

x

which gives no value of T independent of x

∴f(x) can not be periodic

now

say g(x)=cos

2

x+cos

x

which is sum of a

periodic and non-periodic function and such function have no period.

So, cos

x

+cos

2

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