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Introduction to Derivatives · Ages 17–18

Calculus Practice: Intervals of Increase, Decrease, and Extrema

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Designed for 12th-grade calculus students. Differentiate each function, find all critical points, determine the intervals of increase and decrease, and state any local extrema as ordered pairs (x, y).

  1. Find the critical points of f(x) = x^2 - 8x + 12.

  2. Determine the intervals of increase and decrease for f(x) = x^2 - 8x + 12.

  3. Identify and classify the extremum of f(x) = x^2 - 8x + 12 as a coordinate pair.

  4. Find the critical points and intervals of increase and decrease for f(x) = -x^2 + 10x - 7.

  5. Find the local extrema of f(x) = -x^2 + 10x - 7.

  6. Find the intervals of increase and decrease for f(x) = 2x^3 - 6x.

  7. Classify all local extrema for f(x) = 2x^3 - 6x.

  8. Analyze f(x) = x^3 - 3x^2 + 3x - 1 for local extrema. Does x = 1 produce an extremum?

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Calculus Practice: Intervals of Increase, Decrease, and Extrema

Answer key

  1. Find the critical points of f(x) = x^2 - 8x + 12.

    f'(x) = 2x - 8. Setting 2x - 8 = 0 gives x = 4.

  2. Determine the intervals of increase and decrease for f(x) = x^2 - 8x + 12.

    Decreasing on (-infinity, 4); increasing on (4, infinity).

  3. Identify and classify the extremum of f(x) = x^2 - 8x + 12 as a coordinate pair.

    Local minimum at (4, -4).

  4. Find the critical points and intervals of increase and decrease for f(x) = -x^2 + 10x - 7.

    f'(x) = -2x + 10. Critical point: x = 5. Increasing on (-infinity, 5); decreasing on (5, infinity).

  5. Find the local extrema of f(x) = -x^2 + 10x - 7.

    Local maximum at (5, 18).

  6. Find the intervals of increase and decrease for f(x) = 2x^3 - 6x.

    f'(x) = 6x^2 - 6 = 6(x - 1)(x + 1). Critical points: x = -1, x = 1. Increasing on (-infinity, -1) and (1, infinity); decreasing on (-1, 1).

  7. Classify all local extrema for f(x) = 2x^3 - 6x.

    Local maximum at (-1, 4); local minimum at (1, -4).

  8. Analyze f(x) = x^3 - 3x^2 + 3x - 1 for local extrema. Does x = 1 produce an extremum?

    f'(x) = 3(x - 1)^2. Critical point: x = 1. The derivative is positive on both sides of x = 1, so the function increases everywhere on (-infinity, infinity) except at x = 1 where it levels off. There are no local extrema.