Mathematics

The key insight is that periodicity is about repeating behavior over a fixed interval—but products and sums of periodic and non-periodic functions often destroy that pattern. For Statement I, even though $\sin x$ is periodic, multiplying by the non-periodic factor $x$ stretches the oscillations unpredictably, and the derivative $x\cos x + \sin x$ inherits this non-repeating nature. Statement II asks whether this is always true: does differentiating a non-periodic function guarantee a non-periodic result? The answer hinges on finding a counterexample—a non-periodic function whose derivative is periodic.

The key insight is that periodicity is about repeating behavior over a fixed interval—but products and sums of periodic and non-periodic functions often destroy that pattern. For Statement I, even though $\sin x$ is periodic, multiplying by the non-periodic factor $x$ stretches the oscillations unpredictably, and the derivative $x\cos x + \sin x$ inherits this non-repeating nature. Statement II asks whether this is always true: does differentiating a non-periodic function guarantee a non-periodic result? The answer hinges on finding a counterexample—a non-periodic function whose derivative is periodic.