Curve Sketching Math Example 3

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

Example 3

easy
For f(x)=x4−4x3f(x) = x^4 - 4x^3, find and classify all critical points.

Solution

  1. 1
    f′(x)=4x2(x−3)f'(x) = 4x^2(x-3). Critical: x=0x=0 (no sign change), x=3x=3 (sign −- to ++).
  2. 2
    Local min at (3,−27)(3,-27); x=0x=0 is not an extremum.

Answer

Local minimum at (3,−27)(3,-27); x=0x=0 is not an extremum.
When f′′(c)=0f''(c)=0, use the first derivative test. At x=0x=0, f′f' doesn't change sign due to the x2x^2 factor.

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 1 easy

Sketch [formula]: find critical points, monotonicity, concavity, and inflection points.

Example 2 hard

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

Example 4 medium

Find the inflection points of [formula].

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