Module 5 Quiz: Applications of Derivatives II
Quiz Instructions
Answer each question, then reveal the answer. Aim for at least 7 out of 10 correct.
1. On what intervals is f(x) = x4 − 8x² increasing?
f'(x) = 4x³ − 16x = 4x(x−2)(x+2). Increasing on (−2, 0) and (2, ∞).
2. Classify x = 0 as a critical point of f(x) = x4.
f'(0) = 0, f''(0) = 0 (inconclusive). FDT: f' = 4x³ changes − to +. Local minimum.
3. Where is f(x) = x³ − 6x² + 12x concave up?
f''(x) = 6x − 12 > 0 when x > 2. Concave up on (2, ∞).
4. Find the inflection point(s) of f(x) = x3 + 3x² − 1.
f''(x) = 6x + 6 = 0 at x = −1. Sign changes. Inflection point at (−1, 1).
5. Find the horizontal asymptote(s) of f(x) = (2x² + 1)/(x² − 4).
Same degree. Ratio of leading coefficients: y = 2.
6. Two positive numbers sum to 30. What minimum sum of their squares?
S = x² + (30 − x)². S' = 4x − 60 = 0, x = 15. Min sum = 225 + 225 = 450.
7. For curve sketching, what does f''(x) = 0 with a sign change indicate?
An inflection point where the concavity changes.
8. A 60-m fence: three sides of a rectangle (fourth side is a building). Max area?
2x + y = 60, A = x(60 − 2x). A' = 60 − 4x = 0, x = 15, y = 30. Max area = 450 m².
9. Find the local extrema of f(x) = x ln x for x > 0.
f'(x) = ln x + 1 = 0 at x = 1/e. f''(x) = 1/x, f''(1/e) = e > 0. Local min at (1/e, −1/e).
10. True or false: An inflection point can be a local extremum.
False. At an inflection point, the concavity changes, which is incompatible with a local extremum (where the function turns around).