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Calculus I / Applications of Differentiation

Curve Sketching: Concavity, Inflection Points, and the Derivative Tests

An option's value is convex in the underlying price — the P&L of a long option position bends upward relative to a linear hedge. That bending is not captured by the first derivative (delta) alone; it requires the second derivative (gamma). The mathematical framework behind this geometric distinction is concavity: a function that lies above its tangent lines everywhere is concave up, and one that lies below is concave down. The second derivative measures the rate at which the slope is changing, and its sign determines which way the curve bends. Points where the direction of bending switches are inflection points — on a cumulative-returns curve, an inflection point marks where the rate of return growth either peaks or troughs.

This note pulls together the full picture: the Increasing/Decreasing Test from the first derivative, the Concavity Test from the second derivative, both the First and Second Derivative Tests for classifying critical points, and the eight-step curve-sketching algorithm that synthesises all of this into a complete graph.

Learning objectives. By the end of this note you should be able to:
  • apply the Increasing/Decreasing Test to determine where ff rises and falls from the sign of f′f';
  • define concavity precisely and apply the Concavity Test using the sign of f′′f'';
  • locate inflection points and verify that concavity genuinely changes there;
  • classify critical points using both the First Derivative Test (sign change of f′f') and the Second Derivative Test (f′′≠0f'' \ne 0);
  • execute the eight-step curve-sketching algorithm to produce an accurate hand sketch of a function;
  • read the financial meaning of concavity and inflection points in option P&L and return profiles.

In this lesson

  1. Increasing and decreasing: what f′f' says
  2. Classifying critical points: the First Derivative Test
  3. Concavity: what f′′f'' says
  4. Inflection points
  5. The Second Derivative Test
  6. The eight-step curve-sketching algorithm
  7. Slant asymptotes

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