Question

Q. Which of the following function is periodic
a. sin x+sin √2x
b. sin 2x+cos 3x
c. e^xsin x
d. xsin x+cos x

Answer 1

A function f: R → R is said to be periodic if there exists a number T > 0 such that f(x+T) = f(x) for all x ∈ R.

If there is a periodic function$f(x)$   with period T, the function $g(x) = f(kx)$   has of period:

$$T^{\prime}=\frac T k$$

a. sin x+sin √2x  is not periodic.

$y= \sin (x)$ is a periodic function with period $2\pi$.

$y= \sin (\sqrt{2}x)$ is a periodic function with period $2\pi/\sqrt{2}=\sqrt{2}\pi$  

There is no integer, whose division by $2\pi$ and $\sqrt{2}\pi$ will give an integer.

Therefore, the function y = sin x+sin √2x is not periodic.

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b. sin 2x+cos 3x  is a periodic function with period $6\pi$

$y= \sin (2x)$ is a periodic function with period $\pi$ $(\frac {2\pi} {2}= \pi)$ .

$y= \cos (3x)$ is a periodic function with period $(\frac {2\pi} {3}= 2/3 \pi)$ .

$y= \sin (2x)+\cos (3x)$ is a periodic function with period $6\pi.$

(the period is equal to the smallest number, with division of which by $\pi$ and $2/3 \pi$ we get integer numbers).

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c. e^x sin x  is not periodic.

y=sin(x) is a periodic function with period $2\pi.$

y= e^x is not periodic.

The product of functions e^x and sin x will not be a periodic function.

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d. xsin x+cos x is not periodic.

y=xsin x is not periodic. (y=x -  is not periodic, y=sin x -  is periodic $2\pi.$ )

y=cos x is a  periodic function with period $2\pi.$ . .

y= xsin x+cos x  is not periodic. (The sum of the non-periodic (y=xsin x ) and periodic (y=cos x) functions is not a periodic function).