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Calculus practice

Concavity worksheets

Find exact polynomial inflection coordinates and complete concave-up and concave-down intervals with second-derivative sign analysis.

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Find f'', solve f''(x) = 0, build a complete sign chart, and report only actual inflection points with concavity intervals.

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Use exact signs, not sampled decimals

The worked key keeps every derivative zero, analyzes every open interval, and reports exact coordinates with fractions and radicals.

Learning objective

What this activity builds

Solve f''(x) = 0 completely, verify actual sign changes, and evaluate every true inflection input in the original polynomial.

Worked example

Analyze f(x) = x^4.

Concave up on all real numbers; no inflection point.

f''(x) = 12x^2 is zero at 0 but positive on both sides, so concavity does not change.

Teaching notes

Make the practice count

  • Treat zeros of f'' as candidates until signs are checked.
  • Use the original f to find each y-coordinate.
  • Merge concavity intervals across a repeated zero when the sign stays the same.

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Math SheetNameScoreTeacherDateConcavityFind the actual inflection coordinates and give the maximal open intervals where the polynomial is concave up and concave down.1. Find actual inflection points; give concave-up/down intervals.f(x) =2·x2+42. Find actual inflection points; give concave-up/down intervals.f(x) =-2·x3-12·x2-18·x-33. Find actual inflection points; give concave-up/down intervals.f(x) =-2·x4-8·x3+16·x+134. Find actual inflection points; give concave-up/down intervals.f(x) =x2-4·xAn inflection requires a concavity change. -inf/+inf mean infinity.math-sheet.comPage 1 of 2
Math SheetNameScoreTeacherDateConcavityFind the actual inflection coordinates and give the maximal open intervals where the polynomial is concave up and concave down.5. Find actual inflection points; give concave-up/down intervals.f(x) =-x3+2·x-16. Find actual inflection points; give concave-up/down intervals.f(x) =-2·x4+8·x3-4·x2-8·x+37. Find actual inflection points; give concave-up/down intervals.f(x) =-x2+4·x-18. Find actual inflection points; give concave-up/down intervals.f(x) =-2·x3+2An inflection requires a concavity change. -inf/+inf mean infinity.math-sheet.comPage 2 of 2

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