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Quiz: Derivative Tests and Concavity

Test your understanding of derivative tests and concavity with these review questions.

1. According to the First Derivative Test, a local maximum occurs at c if:

  1. f'(c) > 0
  2. f' changes from positive to negative at c
  3. f' changes from negative to positive at c
  4. f''(c) > 0
Show Answer

The correct answer is B. The First Derivative Test says: if f' changes from positive to negative at c, then f has a local maximum at c. The function goes from increasing to decreasing.

Concept Tested: First Derivative Test

2. A function is increasing on an interval when:

  1. f'(x) < 0 throughout the interval
  2. f'(x) > 0 throughout the interval
  3. f''(x) > 0 throughout the interval
  4. f(x) > 0 throughout the interval
Show Answer

The correct answer is B. A function is increasing where its derivative is positive. The positive derivative means the function values are getting larger as x increases.

Concept Tested: Increasing Function

3. A function is concave up on an interval when:

  1. f'(x) > 0
  2. f'(x) < 0
  3. f''(x) > 0
  4. f''(x) < 0
Show Answer

The correct answer is C. A function is concave up where f''(x) > 0. The graph curves upward like a smile, and tangent lines lie below the curve.

Concept Tested: Concave Up

4. An inflection point occurs where:

  1. f'(x) = 0
  2. f''(x) = 0 or undefined, AND concavity changes
  3. f(x) = 0
  4. The function has a maximum
Show Answer

The correct answer is B. An inflection point requires both that f''(x) = 0 or is undefined AND that concavity actually changes (from up to down or down to up) at that point.

Concept Tested: Inflection Point

5. According to the Second Derivative Test, if f'(c) = 0 and f''(c) > 0, then:

  1. f has a local maximum at c
  2. f has a local minimum at c
  3. f has an inflection point at c
  4. The test is inconclusive
Show Answer

The correct answer is B. The Second Derivative Test: if f'(c) = 0 and f''(c) > 0, then f has a local minimum at c. Positive second derivative means concave up (like a bowl).

Concept Tested: Second Derivative Test

6. For f(x) = x³ − 3x, find the interval(s) where f is decreasing.

  1. (−∞, −1)
  2. (−1, 1)
  3. (1, ∞)
  4. (−∞, 0)
Show Answer

The correct answer is B. f'(x) = 3x² − 3 = 3(x−1)(x+1). f' < 0 when −1 < x < 1, so f is decreasing on (−1, 1).

Concept Tested: Decreasing Function

7. A sign chart is used to:

  1. Find where f(x) = 0
  2. Determine where f' is positive or negative
  3. Calculate the derivative
  4. Graph the function exactly
Show Answer

The correct answer is B. A sign chart shows where f' (or f'') is positive, negative, or zero across the domain. This reveals intervals of increase/decrease or concavity.

Concept Tested: Sign Chart

8. For f(x) = x⁴ − 4x³, find the inflection points.

  1. x = 0 only
  2. x = 2 only
  3. x = 0 and x = 2
  4. x = 0, x = 2, and x = 3
Show Answer

The correct answer is C. f'(x) = 4x³ − 12x², f''(x) = 12x² − 24x = 12x(x − 2). f'' = 0 at x = 0 and x = 2. Check that concavity changes at both points.

Concept Tested: Point of Inflection

9. If f''(c) = 0 in the Second Derivative Test, then:

  1. f has a local maximum at c
  2. f has a local minimum at c
  3. f has an inflection point at c
  4. The test is inconclusive; use First Derivative Test
Show Answer

The correct answer is D. When f''(c) = 0, the Second Derivative Test fails (inconclusive). You must use the First Derivative Test or examine the sign chart to classify the critical point.

Concept Tested: Second Derivative Test

10. A function that is both increasing and concave up has:

  1. f' > 0 and f'' < 0
  2. f' > 0 and f'' > 0
  3. f' < 0 and f'' > 0
  4. f' < 0 and f'' < 0
Show Answer

The correct answer is B. Increasing means f' > 0, and concave up means f'' > 0. The function rises while curving upward—it's getting steeper.

Concept Tested: Concavity