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Concavity describes whether a curve bends upward (concave up) or downward (concave down). It is determined by the sign of the second derivative: f''(x) > 0 means concave up; f''(x) < 0 means concave down.
An inflection point is where concavity changes — from concave up to down or vice versa. At an inflection point, f''(x) = 0 or f''(x) is undefined, and f'' must change sign at that point.
Concavity is central to curve sketching and the second derivative test for classifying critical points: a local minimum sits in a concave-up region; a local maximum in a concave-down region.
Real-world interpretation: concave up means the rate of change is increasing (accelerating); concave down means the rate of change is decreasing (decelerating). Population growth slowing down is a concave-down phase.
f''(x) > 0 → concave up | f''(x) < 0 → concave down | f''(c) = 0 + sign change → inflection
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Concavity Rules
f''(x) > 0 on interval → Concave Up (bowl ∪)
f''(x) < 0 on interval → Concave Down (arch ∩)
f''(c) = 0 + sign change → Inflection Point
f''(c) = 0 but no sign change → NOT an inflection point
This calculator is for educational purposes only and does not constitute professional advice. Results are based on standard mathematical formulas. Always verify critical calculations with a qualified professional before making important decisions.