Concavity:

• f''(x) > 0 on interval I: f is concave upward on I (cup, holds water)
• f''(x) < 0 on interval I: f is concave downward on I (cap, sheds water)

Point of Inflection: A point where f'' changes sign (concavity changes). At a point of inflection x = c:

• f''(c) = 0 or f''(c) DNE
• f'' changes sign across c (this is the key condition)

Important: f''(c) = 0 does NOT automatically mean c is an inflection point. Example: f(x) = $x^4$ has f''(0) = 0, but f'' does not change sign (f'' > 0 on both sides of 0). So x = 0 is NOT an inflection point.

For the inflection point, VERIFY sign change of f''.

JEE Application: The point of inflection of a cubic $ax^3$ + $bx^2$ + cx + d is always at x = -$\frac{b}{3a}$, which is the midpoint of the two critical points.