Second Derivative: concavity and inflection points

Second Derivative: Concavity and Inflection Points

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• \(f’’(x) > 0\) → concave up (smiley 😊)
• \(f’’(x) < 0\) → concave down (frowny 😞)
• \(f’’(x) = 0\) with sign change → inflection point (concavity changes)

Classifying Stationary Points

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• \(f’(x_0) = 0\) and \(f’’(x_0) > 0\) → minimum
• \(f’(x_0) = 0\) and \(f’’(x_0) < 0\) → maximum

Example: \(f(x) = x^3 - 3x\)

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\(f’’(x) = 6x\). \(f’’=0\) at \(x=0\). Changes sign → inflection point at \((0,0)\).

📝 Analogy: Position = \(f\), velocity = \(f’\), acceleration = \(f’’\). If \(f’’ > 0\), you’re accelerating; if \(f’’ < 0\), you’re braking.