AHL 5.10 — Second Derivative, Concavity, and the Second Derivative Test

Concept Explanation & Purpose
Second derivative f”(x) Measures how fast the gradient of f(x) is changing.
Used to analyse curvature, acceleration in motion, inflexions, and stability of critical points.
Concavity Concave-up (f”(x) > 0) means the slope is increasing;
Concave-down (f”(x) < 0) means the slope is decreasing. Concavity determines the overall “shape” of the graph, not just local turning.
Second derivative test Determines whether a stationary point is a local maximum, minimum, or inconclusive.
Based on evaluating f”(x) at points where f'(x) = 0.
Point of inflexion A point where f”(x) = 0 AND the concavity changes.
Must check the sign change of f”(x) to confirm.

📌 What the Second Derivative Really Represents

The second derivative describes how the rate of change itself is changing.
While the first derivative answers “how steep?”, the second derivative answers “how is the steepness evolving?”
This is essential for identifying deeper behaviour such as stability, curvature, and acceleration.

  • f”(x) > 0 — Concave-up:
    The slope is increasing; the graph forms a “cup” shape.
    Physically, this represents positive acceleration or increasing marginal returns.
  • f”(x) < 0 — Concave-down:
    The slope is decreasing; the graph forms a “cap” shape.
    In motion, this is deceleration; in economics, diminishing returns.
  • f”(x) = 0 alone is meaningless.
    It could be a maximum, minimum, inflexion, or a flat region with no curvature change.
  • Curvature describes global behaviour.
    Concavity helps predict long-term behaviour, not just one point of the graph.

🌍 Real-World Connection

  • Used in physics to analyse simple harmonic motion (acceleration proportional to negative displacement).
  • Applied in economics to detect increasing vs decreasing marginal returns in production.
  • Used in biology for modelling population growth concavity (logistic growth curve inflexion).
  • In engineering, concavity helps identify stress/strain behaviours under varying loads.

📌 The Second Derivative Test

Once you find a stationary point by solving f'(x) = 0, you evaluate the second derivative:

  • If f”(x₀) > 0 → Local Minimum:
    The function is curving upward; the point sits in a valley.
  • If f”(x₀) < 0 → Local Maximum:
    The function is curving downward; the point sits on a peak.
  • If f”(x₀) = 0 → Inconclusive:
    Could be max/min/inflexion — must check the sign change of f'(x) or f”(x).
  • IB exam tip:
    Always justify your conclusion with “Since f”(x₀) is positive/negative…”.

Second Derivative Test

🧠 Examiner Tip

  • Always state f'(x)=0 first before applying f”(x).
  • Explicitly mention the sign of f”(x) in your classification (IB gives method marks for this).
  • If f”(x)=0, write: “Test inconclusive — checking concavity.”
  • Graphs are heavily rewarded; include them when possible.

📱 GDC Tip

  • Use nDeriv() or graph derivative mode to confirm f'(x)=0 visually.
  • Use a second nDeriv() to evaluate f”(x) at the stationary point.
  • Zoom in around the stationary point to inspect local curvature behaviour.
  • GDC is ideal for verifying sign changes in f”(x) around suspected inflexion points.

📌 Points of Inflexion (More Detailed)

A point of inflexion occurs where the graph changes its concavity.
It often appears where f”(x)=0, but the sign change is the key requirement.

  • If f”(x) changes from positive→negative → concave-up to concave-down.
  • If f”(x) changes from negative→positive → concave-down to concave-up.
  • Common in cubic polynomials, exponential logistics, and S-shaped curves.
  • Failure to check sign change is one of the most frequent causes of lost marks in IB exams.

📐 IA Spotlight

  • Inflexion analysis is valuable when modelling logistic growth or S-curves.
  • Second derivative behaviour strengthens justification of model suitability (Criterion E).
  • Turning points + concavity help validate predictions from fitted functions.
  • Using GDC derivative tools improves accuracy and reduces algebraic error risk.

🔍 TOK Perspective

  • Mathematics models “curvature” in music (vibrations, pitch variation, wave patterns).
  • Does describing music with derivatives make music mathematical, or math musical?
  • Second derivatives reveal structure in nature — does structure imply meaning?
  • How does abstract curvature gain real-world relevance through interpretation?