Inspired by Sandeep Bhardwaj

Let $h(x) = f(x) + g(x)$ .

Counterexample 1: Let $f(x) = |\sin x|$ , $g(x) = |\cos x|$ .

Fundamental period of $f$ is $\pi$ and fundamental period of $g$ is also $\pi$ . However fundamental period of $h = \frac{\pi }{2}< \pi$ , therefore the statement is false.

Counterexample 2: Let $f(x) =$ any periodic function with fundamental period say $t$ . Then let $g(x) = -f(x) +C ~ \forall ~ x$ and some constant $C$ . $f$ and $g$ will have the same fundamental period $t$ , however $h$ will be a constant function. In this case, $h$ has no fundamental period, so the statement is false.

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Suppose fundamental period of $f$ and $g$ is $P$ . Then $P$ is the minimum possible value such that $f(x) = f(x+P)$ and $g(x) = g(x+P)$ . Adding these two equations, we can also say that $h(x) = h(x+P)$ . Therefore $h$ is periodic with period $P$ , so the fundamental period of $h$ is definitely not greater than $P$ .

We can make three cases:

• Case 1: $h$ does not have a fundamental period. This case occurs when $f(x) = -g(x) + C ~ \forall ~ x$ .

• Case 2: Fundamental period of $h$ is half of fundamental period of $f$ and $g$ .

  • If fundamental period of $f$ and $g$ is $P$ and there exists $0 < p < P$ such that $f(x + p) = g(x)$ and $g(x + p) = f(x) ~ \forall ~ x$ , then the fundamental period of $h$ becomes $p$ . This is because $h(x + p) = f(x + p) + g(x + p) = g(x) + f(x) = h(x)$ . This case was illustrated in counterexample 1.

  • We can further show that the only possible value of $p$ is $\frac{P}{2}$ . From the above equations, we obtain $f(x) = f(x + 2p)$ where $p$ takes the minimum value possible. We also know the $f(x) = f(x + P)$ where $P$ is minimum. Hence $P = 2p$ .

• Case 3: Fundamental period of $h$ is equal to fundamental period of $f$ and $g$

As we saw in the examples 1 and 2, case 3 is not always true, sometimes case 2 can be true as well, so the answer is "False, it could have a smaller fundamental period". $_\square$

If f(x) and g(x) are periodic functions with period A and B respectively ,then h(x) =f(x)+g(x) has period as : a. L.C.M. of {A,B} ; if f(x) and g(x) cannot be interchanged by adding a least positive number less than the L.C.M. Of {A,B} . b. K; if f(x) and g(x) can be interchanged by adding a least possitive number k(k<L .C.M. of {A,B} ) These 2 points can be used to perdict the answer .