Applications of Derivatives Quiz

Grade 12 · Math · Quiz

10 questions on critical points, concavity, inflection points, and optimization.

Questions

1. Where is f(x) = x² - 4x + 3 decreasing?

   1. x > 2
   2. x < 2
   3. x > 4
   4. x < 0
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   Answer: x < 2

   f′(x) = 2x - 4 < 0 when x < 2.

2. Find all critical points of f(x) = 2x³ - 6x.

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   f′(x) = 6x² - 6 = 0 → x² = 1 → x = ±1.

3. If f′(x) changes from negative to positive at x = c, then x = c is a:

   1. Local maximum
   2. Inflection point
   3. Local minimum
   4. Critical point only
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   Answer: Local minimum

   First Derivative Test: f′ goes from − to + means the function was decreasing then increasing — a local minimum.

4. A function can have a local maximum at a point where f′ is undefined.

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   For example, f(x) = -|x| has a local maximum at x = 0 where f′(0) is undefined.

5. For f(x) = x⁴, what does the Second Derivative Test tell us about x = 0?

   1. Local max
   2. Local min
   3. Inflection point
   4. Inconclusive
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   Answer: Inconclusive

   f″(x) = 12x². f″(0) = 0, so the Second Derivative Test is inconclusive. (It is actually a global minimum.)

6. Find the x-coordinate of the inflection point of f(x) = x³ - 3x².

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   f″(x) = 6x - 6 = 0 → x = 1. Sign changes at x = 1, confirming inflection.

7. A box with no top has a square base of side x and height h. Volume = 32. To minimize surface area, what equation relates x and h?

   1. x² + 4xh = 32
   2. x²h = 32
   3. 2x² + 4xh = 32
   4. 4xh = 32
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   Answer: x²h = 32

   Volume constraint: V = x²h = 32.

8. Every local maximum is also a global maximum.

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   A function can have multiple local maxima; the global maximum is the highest among all of them.

9. The graph of f is concave up on (1, 5) and concave down on (5, ∞). Where is the inflection point?

   1. x = 1
   2. x = 3
   3. x = 5
   4. x = ∞
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   Answer: x = 5

   Concavity changes at x = 5 — that is the inflection point.

10. What does it mean for f to be increasing on an interval?

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    A positive first derivative means the function's output increases as x increases.

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