Final answer:
To prove that f(x)=(16sinx/4+cosx)-x is strictly decreasing between (π/2, π), we find the derivative f'(x) and show it is negative in this interval. Upon calculation, f'(x) is indeed negative, confirming that f(x) is strictly decreasing.
Step-by-step explanation:
To show that the function f(x) = (16 sin x/4 + cos x) - x is strictly decreasing on the interval (π/2, π), we need to investigate the derivative, f'(x), of the function. If f'(x) is less than 0 for all x in the interval, then f(x) is strictly decreasing over that interval.
First, let's find the derivative of f(x):
'(x) = (16/4) cos(x/4) - sin x - 1
Now, we will determine the sign of f'(x) within the interval (π/2, π).
In that interval, cos(x/4) is positive, and both sin x and 1 are positive as well. Since we are subtracting a positive sine and 1 from a smaller positive cosine term, f'(x) will be negative. Therefore, f(x) is strictly decreasing on the interval (π/2, π).