Practice question · Multiple choice
If f''(c) = 0, the point c need not be an inflection point - x⁴ at x = 0 is the standard counterexample. Why is a vanishing second derivative not enough?
Hints
- Compute f'' for x⁴ and check its sign just left and just right of zero.
- What does "inflection" actually assert about the bending of the curve?
Show the answer
A. Because concavity must change sign, and for x⁴ it does not.
Why
For x⁴ the second derivative is 12x², zero at the origin and positive on both sides, so the curve never changes its bending. An inflection needs a sign change, as x³ supplies. It is exactly parallel to the first-derivative case, f'(c) = 0 does not give an extremum, since x³ has a horizontal tangent and neither. Zeros locate candidates; sign changes confirm them.
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