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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.

  • 2 worked examples
  • 15 in Applications of Derivatives

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, ∞).

1 more worked examplePremium

Find where f(x) = x⁴ - 6x² is concave up and concave down.

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See all 15 Applications of Derivatives formulas on one printable sheet - the same statements and conditions as these pages, typeset and laid out for paper.

Common questions

What is the Concavity?

The second derivative determines concavity. Concave up means the curve opens upward (bowl shape). Concave down means it opens downward. It is one of the applications of derivatives formulas in the CalcRef reference.

How do you use the Concavity?

Worked example. Determine the concavity of f(x) = x³ on (-∞, 0) and (0, ∞). f''(x) = 6x. x < 0: f''(x) < 0 → concave down. x > 0: f''(x) > 0 → concave up. Answer: Concave down on (-∞, 0), concave up on (0, ∞).

Practice

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