First derivative test
Check the sign of f' on each side of a critical point. Positive then negative means a local maximum. Negative then positive means a local minimum.
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Analogy
Walking over a hill: uphill, then downhill, means you passed a peak.
Example
x < -1: f' > 0, rising.-1 < x < 1: f' < 0, falling.x > 1: f' > 0, rising.Max at x = -1, min at x = 1.
In practice
A reliable way to classify critical points without needing the second derivative.
Practice
Flashcards
- What does a sign change of f' from positive to negative mean?A local maximum.
Interview questions
- Find and classify the critical points of f(x) = x^3 - 3x.
- f'(x) = 3x^2 - 3 = 0 gives x = -1 and 1
- f''(x) = 6x
- f''(-1) < 0 so local maximum, value 2
- f''(1) > 0 so local minimum, value -2
Questions people ask
What if f' does not change sign at a critical point?
Then it is neither a maximum nor a minimum. The function keeps going in the same direction. For f(x) = x cubed, f' is positive on both sides of 0, so x = 0 is a flat spot on the way up, not a peak or valley.
How do I check the sign of f' between critical points?
Pick one test value in each interval and put it into f'. Only the sign matters. For 3x squared - 3, the value x = 0 gives -3, so f' is negative between -1 and 1. A continuous f' can only change sign where it is zero.