Home › Courses › Calculus I Foundations (MATH 151) › Applications: Extrema & Curve Sketching

Calculus I Foundations (MATH 151) · Module 5 of 11

Applications: Extrema & Curve Sketching

Using a function's derivatives to find its peaks, valleys, and overall shape.

In this module
  1. Critical Points
  2. The First Derivative Test
  3. The Second Derivative Test & Concavity
  4. Sketching Curves
  5. Check yourself

Critical Points

  • A critical point occurs where f'(x) = 0 or f'(x) is undefined; critical points are candidates for local maxima or minima.
Worked example

Find the critical points of f(x) = x³ − 3x.

Solution. f'(x) = 3x² − 3 = 0 → x² = 1 → x = ±1. These are the critical points.

The First Derivative Test

  • The first derivative test classifies a critical point by checking the sign of f' on either side: positive to negative means a local max; negative to positive means a local min; no change means neither.
Critical points of f(x) = x³ − 3x
xylocal max (−1, 2)local min (1, −2)
f′(x) = 3x² − 3 = 0 at x = ±1. f′ changes from + to − at x = −1 (local max, f = 2) and from − to + at x = 1 (local min, f = −2).
Worked example

Using the first derivative test, is x = 1 a local max or min for f(x) = x³ − 3x, where f'(x) = 3x² − 3?

Solution. For x slightly less than 1 (e.g. 0.9), f'(0.9) < 0; for x slightly more than 1 (e.g. 1.1), f'(1.1) > 0. Since f' goes from negative to positive, x = 1 is a local minimum.

The Second Derivative Test & Concavity

  • The second derivative test: if f'(c) = 0 and f''(c) > 0, c is a local min (concave up); if f''(c) < 0, c is a local max (concave down); if f''(c) = 0, the test is inconclusive.
  • An inflection point is where concavity changes, meaning f'' changes sign.
Worked example

If f'(c) = 0 and f''(c) > 0, what can you conclude about x = c?

Solution. x = c is a local minimum, since a positive second derivative means the function is concave up at that point, curving upward around a minimum.

Sketching Curves

  • A full curve sketch uses critical points (from f') to find local extrema, and inflection points (from f'') to find where concavity changes, combined with intercepts and asymptotes, to represent a function's shape accurately.
Worked example

What key pieces of information from f' and f'' help you sketch an accurate graph of f?

Solution. f' identifies critical points and where the function is increasing or decreasing; f'' identifies inflection points and where the function is concave up or down — together they reveal the overall shape, local extrema, and curvature.

Check yourself

2 of this module's 5 quiz questions. Pick your answers, then check them in the app, with an explanation for every question.

1. What defines a critical point?

2. In the first derivative test, what does it mean if f' changes from positive to negative at a critical point?

Check my answers, free