f(x) = sin 2x increasing and decreasing interval
Question
f(x) = sin 2x increasing and decreasing interval
Solution
To find the increasing and decreasing intervals of the function f(x) = sin(2x), we first need to find its derivative.
Step 1: Find the derivative of f(x) = sin(2x) The derivative of sin(2x) is f'(x) = 2cos(2x) using the chain rule.
Step 2: Set the derivative equal to zero and solve for x 0 = 2cos(2x) cos(2x) = 0 2x = π/2 + kπ, where k is an integer. x = π/4 + kπ/2
Step 3: Determine the intervals where the function is increasing or decreasing We test the intervals between the critical points (x = π/4 + kπ/2) in the derivative.
- For x in (-∞, π/4): f'(x) = 2cos(2x) > 0, so f(x) is increasing.
- For x in (π/4, 3π/4): f'(x) = 2cos(2x) < 0, so f(x) is decreasing.
- For x in (3π/4, 5π/4): f'(x) = 2cos(2x) > 0, so f(x) is increasing.
- For x in (5π/4, 7π/4): f'(x) = 2cos(2x) < 0, so f(x) is decreasing.
- And so on...
So, the function f(x) = sin(2x) is increasing on the intervals (-∞, π/4 + kπ), where k is an even integer, and decreasing on the intervals (π/4 + kπ, 3π/4 + kπ), where k is an odd integer.
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