Common Mistakes

Frequent Errors

1. Treating a local max as automatically global. 2. Claiming odd-degree polynomials always have both a global max and min. 3. Placing extrema at the zeros instead of between zeros. 4. Confusing inflection points with local extrema. 5. Forgetting that restricted-domain endpoints can also be local extrema. 6. Misreading degree from a non-standard form (need equivalent polynomial form). 7. Saying "rate of change is zero" when they mean "rate of change stops increasing." 8. Ignoring leading-coefficient sign when discussing which global extremum exists for even degree.

Why These Happen

1.4 introduces several nearby ideas—degree, turns, global vs local, inflection—so students blend them. Graphs with multiple bumps invite "highest bump = only max that matters" thinking.

How to Fix Them

• Always ask: "Highest nearby, or highest everywhere?"
• For even degree, first decide global max vs global min from end behavior / leading coefficient.
• Mark zeros, then mark at least one turn between each consecutive pair.
• For inflection: ask whether steepness is still increasing or has started decreasing.

Avoidance Scripts

• "Local = neighborhood. Global = whole graph."
• "Zeros trap extrema between them; zeros are not the turns."
• "Inflection changes concavity; extremum changes increase/decrease."

Quick Self-Check

• Did I specify local vs global?
• Did I use even-degree language only when degree is even?
• Did I separate turning points from inflection points?