FKM Kompass
L2

Curve sketching with derivatives

The first derivative's sign locates where a function rises or falls and where it has local extrema; the second derivative's sign locates where the graph is convex or concave and where it has inflection points — together they turn an algebraic expression into an accurate sketch without plotting points.

Practise this concept

A function's first and second derivatives are a complete toolkit for sketching its graph without plotting dozens of points. The sign of f'(x) tells you where the curve is rising (f' > 0) or falling (f' < 0); where f'(x) = 0 you have a candidate extremum, but only a genuine sign change of f' across that point confirms a maximum or minimum — a flat spot where the sign doesn't change, like x³ at the origin, is neither.

The second derivative answers a different question: not whether the curve is going up or down, but how it is bending. Where f''(x) > 0 the curve bends upward (convex, holding water); where f''(x) < 0 it bends downward (concave). A point where f''(x) = 0 is only an inflection point if the concavity actually flips sign there — otherwise it is a false alarm, just like a stationary point that isn't an extremum.

Combining both derivatives — rising/falling from f', convex/concave from f'' — lets you sketch a curve's overall shape (its "story": where it climbs, where it turns, where it bends) directly from the algebra, which is the whole point of curve sketching as opposed to blind point-plotting.

Key ideas

Requires: The derivative

Common misconceptions

  • Believes every point where f'(x) = 0 is a local extremum, without checking that the derivative's sign actually changes there — missing that f(x) = x³ has a stationary point at x = 0 that is neither a maximum nor a minimum.
  • Confuses the sign of f'(x) (rising or falling) with the sign of f''(x) (convex or concave) — believing a function must be concave wherever it is decreasing.
  • Assumes every point where f''(x) = 0 is an inflection point, without checking that the concavity actually changes sign there.

Resources

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Part of these UT courses

This concept is mapped as covering part of the following course(s) in the University of Tartu curriculum.