How do you prove algebraically whether the following is periodic?

f(x) = 3cos3x + 5sin4x

The sum of two periodic functions is always periodic. In your case, note than when x is a multiple of 2 pi, both functions repeat the values of x = 0.

To prove algebraically whether the function f(x) = 3cos(3x) + 5sin(4x) is periodic, we need to examine the periods of both the cosine and sine functions individually.

The cosine function has a period of 2π. This means that if you add any multiple of 2π to the input of the cosine function, the output will repeat. So cos(3(x + 2π)) = cos(3x + 6π) = cos(3x).

Similarly, the sine function also has a period of 2π. So sin(4(x + 2π)) = sin(4x + 8π) = sin(4x).

Since both the cosine and sine functions are periodic with a period of 2π, we can conclude that the sum of these two functions, which is f(x) = 3cos(3x) + 5sin(4x), will also be periodic.

To determine the period of f(x), we need to find the least common multiple (LCM) of the periods of both the cosine and sine functions:

LCM(2π, 2π) = 2π.

Therefore, the function f(x) = 3cos(3x) + 5sin(4x) is periodic with a period of 2π.