Curve Sketching: Concavity and Inflection Points (Calculus I)
Free printable Calculus I worksheet on concavity, inflection points and the second derivative test, with cubics and quartics whose second derivatives are solved and sign-checked in code so every stated inflection point is a genuine sign change.
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Calculus I · Math worksheet
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Curve Sketching: Concavity and Inflection Points
Find f''(x). f is concave up where f'' > 0 and concave down where f'' < 0; an inflection point is where f'' changes sign. For the second derivative test, evaluate f'' at a critical point: negative means local maximum, positive means local minimum.
- 1.Find the inflection point of f(x) = x³ - 9x² - 4x + 4 and state the intervals on which f is concave up and concave down.
- 2.Find all inflection points of f(x) = x⁴ - 8x³ + 18x² - 3x and state the intervals of concavity.
- 3.f(x) = x³ - 12x - 2 has a critical point at x = -2. Use the second derivative test to classify it.
- 4.Find the inflection point of f(x) = x³ + 6x² - x + 4 and state the intervals on which f is concave up and concave down.
- 5.Find all inflection points of f(x) = x⁴ - 10x³ + 24x² - 4x and state the intervals of concavity.
- 6.f(x) = x³ - 9x² + 15x has a critical point at x = 1. Use the second derivative test to classify it.
- 7.Find the inflection point of f(x) = x³ - 9x² + x - 4 and state the intervals on which f is concave up and concave down.
- 8.Find all inflection points of f(x) = x⁴ - 2x³ - 12x² and state the intervals of concavity.
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