How do you determine where f is increasing or decreasing?If f ′(x) 0 on an interval, then f is increasing on that interval.If f ′(x) 0 on an interval, then f is decreasing on that interval.
Question
How do you determine where f is increasing or decreasing?If f ′(x) 0 on an interval, then f is increasing on that interval.If f ′(x) 0 on an interval, then f is decreasing on that interval.
Solution
To determine where a function f is increasing or decreasing, you need to follow these steps:
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Find the derivative of the function, denoted as f'(x). The derivative represents the slope of the function at any given point.
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Set the derivative equal to zero and solve for x. These solutions are the critical points of the function.
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Use these critical points to divide the number line into intervals.
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Pick a test point from each interval and substitute it into the derivative.
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If f'(x) > 0 at that test point, then f is increasing on that interval. If f'(x) < 0 at that test point, then f is decreasing on that interval.
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Repeat steps 4 and 5 for all intervals.
Remember, if the derivative of a function is positive over an interval, then the function is increasing over that interval. If the derivative is negative over an interval, then the function is decreasing over that interval.
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