Yeah, sure!
1) Since we know that f(x+T) = f(x)
2) and f(x) = pi(x)^2 - cos (x), we can plug in f(x+T) for anywhere there is an f(x) for our manipulated side (left).
Since this is a proof, there are going to be two sides.
3) Then we get pi(x+T)^2 - cos(x+T) = pi(x)^2 - cos(x). Our left side going to be manipulated while our right side is going to be our "constant" with the original expression pi(x)^2 - cos(x).
4) We can expand the expression which gives us pi(x^2+2(x)T+T^2) - cos(x+T) = pi(x)^2 - cos(x)
5) Then we can foil the pi, which gives us pi(x)^2+ 2pi(x)T + pi(T)^2 - cos(x+T) = pi(x)^2 - cos(x)
6) We can then subtract pi(x)^2 from both sides which will leave us with 2pi(T)x + pi(T)^2 - cos(x+T) = -cos(x)
I might've switched the T and the x in some places (such as 2pi(x)T) compared from my previous answer (where it was 2pi(T)x), but it doesn't affect the actual proof. We can still test our proof just fine in this case. Where you arrange the x and T's is mostly whatever is easier to understand. It's multiplication, so it's fine to flip them generally. It's like 4x5 is the same as 5x4.
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I already explained this earlier but here's a more detailed explanation for the contradiction of the proof.
7) Testing the proof:
Now since we are given the hint that f(x) is a periodic function of period T when T > 0 for all x in the domain of f. We can test T = 0. Plug in 0 where ever there's a T and we get -cos(x) = -cos(x). This isn't what we are looking for because this shows that f(x) is a periodic function when T = 0 and is also a valid proof. Not a contradiction.
So since we need to show that f(x) is not a periodic function, we need to make the proof impossible. Since we tried T = 0, we can also test x = 0. Plug in 0 where ever there's an x, and we get pi(T)^2 - cos(T) = -1. This shows that f(x) is not a periodic function when x = 0 and is not a valid proof. It's a contradiction.
Solution: The function is not periodic when x = 0. Unless when T = 0, the function is periodic.
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I actually forgot to show the expanding and foiling part on my actual assignment since I did it on a separate piece of paper. So it looked like I skipped two steps lol.
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