A problem about periodic function (see supplement) Given the function f (x) = sinwx coswx, if there is a real number X1 such that f (x1) ≤ f (x) ≤ f (x1 + 4) holds for any real number x, what is the minimum value of positive number W? The answer is π / 4. I mainly want to know why it is enough to take half a period between the maximum and the minimum. I always feel that for the interval (K π, K π + π / 2), there is exactly half a period, but not any f (x) has a function value corresponding to this interval Why is half a period between F (x1) and f (x1 + 4) enough? Does it have to be the maximum and minimum?

A problem about periodic function (see supplement) Given the function f (x) = sinwx coswx, if there is a real number X1 such that f (x1) ≤ f (x) ≤ f (x1 + 4) holds for any real number x, what is the minimum value of positive number W? The answer is π / 4. I mainly want to know why it is enough to take half a period between the maximum and the minimum. I always feel that for the interval (K π, K π + π / 2), there is exactly half a period, but not any f (x) has a function value corresponding to this interval Why is half a period between F (x1) and f (x1 + 4) enough? Does it have to be the maximum and minimum?

Why is it enough to take half a period between the maximum and the minimum
That's because it takes one cycle from the maximum to the next
f(x)=√2sin(wx-π/4)
There is a real number X1 such that for any real number x,
F (x1) ≤ f (x) ≤ f (x1 + 4) holds
Then f (x1) is the minimum and f (x1 + 4) is the maximum
The distance between X1 + 4 and X1 is KT + T / 2, K ∈ Z
[k-times period plus half period]
That is KT + T / 2 = 4, (2 π / W) (K + 1 / 2) = 4
∴w=π/2(k+1/2)
When k = 0, the minimum value of positive number W is π / 4