Practice question · Sort into groups
Sort each function by whether x = 0 is an inflection point of its graph.
Groups: x = 0 is an inflection point · x = 0 is not an inflection point
- x to the fourth
- x to the fifth
- x cubed
- x squared
- x to the fourth plus x
- x cubed minus x
Hints
- Compute f'' for each and ask whether it CHANGES SIGN at 0, not merely whether it vanishes.
- For x to the fourth, f'' = 12 x squared, which is never negative.
Show the answer
x = 0 is an inflection point: x cubed, x to the fifth, x cubed minus x
x = 0 is not an inflection point: x to the fourth, x squared, x to the fourth plus x
Why
For the even powers f'' is 12x squared or 2, neither of which changes sign, so those graphs stay concave up on both sides. The odd powers give f'' = 6x and 20x cubed, which do switch sign. Adding the linear term x changes the slope but not f'' at all, so item 5 still fails.
Practise Higher Derivatives and Concavity
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