To determine whether the functions are periodic, we need to see if there exists a positive constant T such that f(x+T)=f(x) for all x in the domain of the function. Let's analyze each option one by one. Option A: x+sin‌2x is a periodic function The function x+sin‌2x consists of a linear term x and a periodic term sin‌2x. The linear term x is not periodic because it continuously increases or decreases without repeating any values. Therefore, the sum x+sin‌2x cannot be periodic because the non-periodic linear term dominates the behavior of the function as x increases. Option B: x+sin‌2x is not a periodic function As explained above, the function x+sin‌2x is not periodic due to the presence of the linear term x. Thus, this statement is correct. Option C: cos(√x+1) is a periodic function To determine the periodicity of cos(√x+1), we need to check if there exists a positive constant T such that cos(√x+T+1)=cos(√x+1) for all x. However, the argument of the cosine function, √x+1, is not periodically repeating as x increases. The function √x increases without bound, thus the argument of the cosine function does not repeat its values periodically. Option D:cos(√x+1) is not a periodic function As explained above, the argument √x+1 does not repeat periodically, so cos(√x+1) is not a periodic function. Thus, this statement is correct. Therefore, the correct statements are: Option B: x+sin‌2x is not a periodic function Option D:cos(√x+1) is not a periodic function