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Calculus practice

Critical Points worksheets

Find, classify, and plot exact polynomial critical coordinates, then report complete increasing and decreasing intervals.

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Differentiate, solve f'(x) = 0, build a complete sign chart, and report every critical coordinate and behavior interval.

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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, use derivative signs on every interval, and classify each critical coordinate.

Worked example

Analyze f(x) = x^3.

Critical point (0, 0); increasing on all real numbers.

f'(x) = 3x^2 is zero at 0 but positive on both sides, so the point is stationary without being a maximum or minimum.

Teaching notes

Make the practice count

  • Keep repeated derivative zeros in the sign chart.
  • Substitute each critical input into the original polynomial.
  • Merge behavior intervals when an even-multiplicity zero does not change the derivative sign.

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Choose 6, 8, or 10 problems and any nonempty selection of quadratic, cubic, and quartic polynomials.

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Math SheetNameScoreTeacherDateCritical PointsFind every critical coordinate, classify it, and give the maximal open intervals where the polynomial increases and decreases.1. Find and classify all critical points; give increasing/decreasing intervals.f(x) =x2+4·x+32. Find and classify all critical points; give increasing/decreasing intervals.f(x) =x3-6·x2+9·x-13. Find and classify all critical points; give increasing/decreasing intervals.f(x) =-x4-8·x3-18·x2-8·x+124. Find and classify all critical points; give increasing/decreasing intervals.f(x) =x2-1Use the complete sign chart of f' to classify each point. -inf/+inf mean infinity.math-sheet.comPage 1 of 2
Math SheetNameScoreTeacherDateCritical PointsFind every critical coordinate, classify it, and give the maximal open intervals where the polynomial increases and decreases.5. Find and classify all critical points; give increasing/decreasing intervals.f(x) =-2·x3+4·x-26. Find and classify all critical points; give increasing/decreasing intervals.f(x) =2·x4-8·x3+4·x2+8·x-107. Find and classify all critical points; give increasing/decreasing intervals.f(x) =x2+4·x+78. Find and classify all critical points; give increasing/decreasing intervals.f(x) =-x3+6·x2-12·x+7Use the complete sign chart of f' to classify each point. -inf/+inf mean infinity.math-sheet.comPage 2 of 2

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