Practice question · Match the pairs
Match each derivative condition to what it says about the graph.
- f' is positive
- f'' is positive
- f'' is negative
- f'' changes sign
- The curve is concave up
- The curve is concave down
- There is an inflection point
- The function is rising
Hints
- The first derivative controls direction and the second controls bending.
- Only one of these entries is about a CHANGE rather than a value.
Show the answer
- f' is positive → The function is rising
- f'' is positive → The curve is concave up
- f'' is negative → The curve is concave down
- f'' changes sign → There is an inflection point
Why
Direction comes from f' and bending from f''; the two are independent, so a curve can rise while bending either way. Only the last entry involves a change of sign, which is why inflection points cannot be detected from a single value of f''.
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