Curve Sketching — f, f', f'' Relationships
f'(x) tells you about f:
f' > 0 → f is increasing
f' < 0 → f is decreasing
f' = 0 → potential extremum
f''(x) tells you about concavity:
f'' > 0 → f is concave up (smile shape)
f'' < 0 → f is concave down (frown shape)
f'' = 0 → potential inflection point
Inflection point: Where concavity CHANGES. f'' must change sign, not just equal zero.
AP exam favorite: Given a graph of f'(x), describe the behavior of f(x). Where is f increasing? Where are the local extrema? Where are inflection points?
Key insight: Zeros of f' are extrema of f. Zeros of f'' are inflection points of f. Extrema of f' are inflection points of f.
Reference:
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