Increasing, Decreasing, and Extrema
Study Notes
Concept Overview
To analyze the behavior of a function f (x), we study its derivative f ′ (x):
• If f ′ (x) > 0, the function is increasing.
• If f ′ (x) < 0, the function is decreasing.
• If f ′ (x) changes from positive to negative, f (x) has a relative maximum.
• If f ′ (x) changes from negative to positive, f (x) has a relative minimum.
Critical numbers occur when f ′ (x) = 0 or f ′ (x) is undefined. These points are candidates
for extrema.
Step-by-Step Process
1. Compute f ′ (x).
2. Solve f ′ (x) = 0 or find where f ′ (x) is undefined.
3. Test intervals around critical numbers to determine where f ′ (x) is positive (increasing)
or negative (decreasing).
4. Identify relative maxima and minima based on sign changes.
Example: f (x) = x3 − 3x2 + 12
1. Derivative:
f ′ (x) = 3x2 − 6x = 3x(x − 2).
2. Critical numbers: f ′ (x) = 0 ⇒ x = 0, x = 2.
3. Test intervals:
• For x < 0, f ′ (x) > 0 (increasing).
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Study Notes
Concept Overview
To analyze the behavior of a function f (x), we study its derivative f ′ (x):
• If f ′ (x) > 0, the function is increasing.
• If f ′ (x) < 0, the function is decreasing.
• If f ′ (x) changes from positive to negative, f (x) has a relative maximum.
• If f ′ (x) changes from negative to positive, f (x) has a relative minimum.
Critical numbers occur when f ′ (x) = 0 or f ′ (x) is undefined. These points are candidates
for extrema.
Step-by-Step Process
1. Compute f ′ (x).
2. Solve f ′ (x) = 0 or find where f ′ (x) is undefined.
3. Test intervals around critical numbers to determine where f ′ (x) is positive (increasing)
or negative (decreasing).
4. Identify relative maxima and minima based on sign changes.
Example: f (x) = x3 − 3x2 + 12
1. Derivative:
f ′ (x) = 3x2 − 6x = 3x(x − 2).
2. Critical numbers: f ′ (x) = 0 ⇒ x = 0, x = 2.
3. Test intervals:
• For x < 0, f ′ (x) > 0 (increasing).
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