Curve Sketching Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Sketch f(x)=x3−3x2f(x) = x^3 - 3x^2: find critical points, monotonicity, concavity, and inflection points.

Solution

  1. 1
    f′(x)=3x2−6x=3x(x−2)f'(x) = 3x^2-6x = 3x(x-2). Critical points: x=0,2x=0, 2.
  2. 2
    Increases on (−∞,0)(-\infty,0), decreases on (0,2)(0,2), increases on (2,∞)(2,\infty).
  3. 3
    Local max (0,0)(0,0), local min (2,−4)(2,-4).
  4. 4
    f′′(x)=6x−6f''(x)=6x-6. Zero at x=1x=1; sign changes: inflection at (1,−2)(1,-2).
  5. 5
    Concave down on (−∞,1)(-\infty,1), concave up on (1,∞)(1,\infty).

Answer

Local max (0,0)(0,0); local min (2,−4)(2,-4); inflection (1,−2)(1,-2).
Systematic: find f′f' for critical points/monotonicity, f′′f'' for concavity/inflection. Use sign charts.

About Curve Sketching

Using the first and second derivatives to determine a function's behavior: intervals of increase/decrease, local maxima/minima, concavity (up/down), and inflection points, then combining this information to sketch an accurate graph.

Learn more about Curve Sketching →

More Curve Sketching Examples

Example 2 hard

Sketch [formula]: find domain, asymptotes, critical points, and concavity.

Example 3 easy

For [formula], find and classify all critical points.

Example 4 medium

Find the inflection points of [formula].

← All Curve Sketching Examples Curve Sketching Concept