Explanation:

- Function given is f(x) = sin[n²]x + cos[-p²]x + cos[-p²]. The greatest integer function [.] acts on powers n² and -p².

- Since these are constants, [n²] and [-p²] are integers. Therefore, trigonometric functions will evaluate based on these integer multiples.

- Let’s evaluate each term.

- For sin[n²](p/4), if [n²] is even, full cycles of sin make this sin(0) = 0.

- For cos[-p²](p/4), [-p²] is negative so the angle could be beyond standard range, but cosine is periodic. Hence, it could potentially resolve to cos(0 or p) = ±1.

- For cos[-p²], since it also involves [-p²], similar simplification applies.

- Evaluate the specific expressions for plausible n² values.

- Find that calculation matched expected integer cases giving net operation values.

- Examining options:

- 1 : This suggests total positive unity. Improbable after breakdown.

- 0 : This suggests neutrality. Possible with correct balance.

- -1 : Total negative. Improbable due to setup symmetry.

- 1/v2 (Option 4) : Given configuration, resolve to this as plausible with angle intricacies.

Thus, your calculation with option:4 – 1/v2 seems correct based on reasoning.