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

Increasing on $[0,π]$

Decreasing on $[0,π]$

Increasing on $[0,$\frac{\pi}{2}$)$ , Decreasing on $($\frac{\pi}{2}$,$\pi$]$

Increasing on $($\frac{\pi}{2}$,$\pi$]$ , Decreasing on $[0,$\frac{\pi}{2}$)$

검증된 단계별 안내

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.

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.

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.

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.

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.

11:22m

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Identify the intervals on which the function is increasing or decreasing.f(x)=sin2xf(x)=\(\sin\)^2xf(x)=sin2x on [0,π][0,\(\pi\)][0,π]

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