Finding the Fundamental Period of f(x)

To find the fundamental period of a function, we need to determine the smallest positive value of T such that f(x) = f(x + T) for all x.

Given function: f(x) = (sin 12x)/(1 + cos^2 6x)

Step 1: Determine the period of sin 12x

The period of sin ax is 2π/a. Therefore, the period of sin 12x is 2π/12 = π/6.

Step 2: Determine the period of cos^2 6x

The period of cos^2 ax is π/a. Therefore, the period of cos^2 6x is π/6.

Step 3: Determine the LCM of the periods of sin 12x and cos^2 6x

The LCM of π/6 and π/6 is π/6.

Therefore, the fundamental period of f(x) is π/6.

Explaining the Discrepancy

While it may be true that f(π/12) gives the same value as f(π/6), this does not necessarily mean that the fundamental period of the function is π/12.

The fundamental period refers to the smallest positive value of T such that f(x) = f(x + T) for all x. In this case, we have shown that the fundamental period of f(x) is π/6.

It is possible that f(π/12) and f(π/6) have the same value due to the symmetry of the function. However, this does not change the fact that the fundamental period is π/6.

Conclusion

- The fundamental period of f(x) = (sin 12x)/(1 + cos^2 6x) is π/6.
- While f(π/12) and f(π/6) may have the same value, this does not affect the determination of the fundamental period.
- It is important to understand the definition of fundamental period and apply it appropriately to find the period of a function.