Explanation
1. Simplify the function:
Using the trigonometric identity cos2x=2cos2x−1, we can rewrite the function:
f(x)=cos2x
2. Find the derivative:
To determine where the function is decreasing, we find its derivative f′(x) and set it to be less than zero:
f′(x)=dxd(cos2x)=−2sin2x
For the function to be decreasing, we require f'(x) < 0:
-2\sin 2x < 0 \implies \sin 2x > 0
3. Solve the inequality:
The sine function sinθ is positive in the first and second quadrants, i.e., 0 < \theta < \pi.
Here, θ=2x:
0 < 2x < \pi
0 < x < \frac{\pi}{2}
4. Determine the length of the interval:
The function is decreasing on the interval (0,2π).
The length of this interval is:
Length=2π−0=2π
Final Answer:
The correct option is (c) 2π.
