CalcRef

Applications of Derivatives · Curve Analysis

Concavity

f′′(x)>0⇒concave up;f′′(x)<0⇒concave downf''(x) > 0 \Rightarrow \text{concave up}; \quad f''(x) < 0 \Rightarrow \text{concave down}

The second derivative determines concavity. Concave up means the curve opens upward (bowl shape). Concave down means it opens downward.

Worked examples

Determine the concavity of f(x) = x³ on (-∞, 0) and (0, ∞).
  1. f''(x) = 6x
  2. x < 0: f''(x) < 0 → concave down
  3. x > 0: f''(x) > 0 → concave up

Answer: Concave down on (-∞, 0), concave up on (0, ∞).

Related formulas

Inflection Points

Practice this and 135 more formulas in the CalcRef workspace — quizzes, reference tables, a 16-category unit converter, and an expression evaluator.