Curve Sketching Math Example 4

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Example 4

medium
Find the inflection points of f(x)=x4−6x2f(x) = x^4 - 6x^2.

Solution

  1. 1
    f′′(x)=12x2−12=12(x−1)(x+1)f''(x) = 12x^2-12 = 12(x-1)(x+1).
  2. 2
    Sign changes at x=±1x=\pm1. f(−1)=f(1)=−5f(-1)=f(1)=-5.
  3. 3
    Inflection points: (−1,−5)(-1,-5) and (1,−5)(1,-5).

Answer

Inflection points at (−1,−5)(-1,-5) and (1,−5)(1,-5).
An inflection requires f′′f'' to change sign, not merely equal zero. Both x=±1x=\pm1 qualify.

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 3 easy

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

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