Where the concavity changes: A point of inflexion is where the curve changes concavity — from concave up to concave down, or vice versa. There, f''(x) = 0 and f'' changes sign.
f'' = 0 is necessary, not sufficient: f''(x) = 0 alone is not enough — the second derivative must also change sign through that point for it to be an inflexion.
Solve f''(x) = 0, then check the sign change: Solve f''(x) = 0 for candidate x-values, then confirm f'' changes sign across each. The y-coordinate comes from the original f(x).
IB-style question — find the inflexion point
Find the point of inflexion of f(x) = x³ − 6x² + 5x.
Step by step
- f'(x) = 3x² − 12x + 5, f''(x) = 6x − 12 = 0 ⇒ x = 2.
- f'' goes − (x<2) to + (x>2): sign change ✓. y = f(2) = 8 − 24 + 10.
Final answer
Point of inflexion at (2, −6).
y from the original function: After finding x from f''(x) = 0, get y by substituting into the original f(x).
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f'' = 0 without a sign change is NOT an inflexion: Always check a value of f'' on each side. If f'' has the same sign on both sides (no change), it is not a point of inflexion — for example y = x⁴ at x = 0.
IB-style question — is it an inflexion?
Show that y = x⁴ does NOT have a point of inflexion at x = 0, even though f''(0) = 0.
Step by step
- f''(x) = 12x². Check the sign either side of 0.
- Same sign both sides → no change of concavity.
Final answer
f'' = 0 at x = 0 but f'' > 0 on both sides (no sign change), so there is no point of inflexion.
Always test both sides: The sign change of f'' is what makes it an inflexion — never conclude from f'' = 0 alone.