Practice question · Sort into groups
Each item reports how the sign of f' behaves as x passes through a critical point. Sort them.
Groups: Local maximum · Local minimum · Neither
- f' goes from positive to negative
- f' is negative on both sides
- f' is positive on both sides
- f' goes from negative to positive
Hints
- Picture the function rising and falling, and ask where the turn happens.
- If the function never turns around, it has no extreme value there at all.
Show the answer
Local maximum: f' goes from positive to negative
Local minimum: f' goes from negative to positive
Neither: f' is positive on both sides, f' is negative on both sides
Why
Rising then falling gives a peak; falling then rising gives a valley. If the sign never changes the function simply pauses and continues in the same direction, that is an inflection with a horizontal tangent, and it is neither a maximum nor a minimum.
Practise Applications of Derivatives
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