Identify the intervals on which the function is increasing or decreasing.
$f(x)=\sin^2x$ on $[0,\pi]$

• A. Increasing on $[0,π]$
• B. Decreasing on $[0,π]$
• C. Increasing on $[0,\frac{\pi}{2})$, Decreasing on $(\frac{\pi}{2},\pi]$
• D. Increasing on $(\frac{\pi}{2},\pi]$, Decreasing on $[0,\frac{\pi}{2})$

Answer: (C)

Verified step by step guidance

1. First, understand that to determine where the function f(x) = sin^2(x) is increasing or decreasing, we need to find its derivative, f'(x). This will help us identify the critical points and intervals of increase or decrease.
2. Calculate the derivative of f(x) = sin^2(x). Use the chain rule: if f(x) = (sin(x))^2, then f'(x) = 2 * sin(x) * cos(x). This simplifies to f'(x) = sin(2x) using the double angle identity.
3. Set the derivative f'(x) = sin(2x) equal to zero to find the critical points. Solve sin(2x) = 0 for x in the interval [0, π]. The solutions are x = 0, π/2, and π.
4. Analyze the sign of f'(x) = sin(2x) in the intervals determined by the critical points: [0, π/2), (π/2, π]. If f'(x) > 0, the function is increasing; if f'(x) < 0, the function is decreasing.
5. Conclude the intervals: f(x) is increasing on [0, π/2) because sin(2x) > 0, and decreasing on (π/2, π] because sin(2x) < 0.