Topic Review — Further Applications of Differentiation

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This review covers all three lessons in Further Applications of Differentiation: second derivative, concavity and inflection points; second derivative test for stationary points; and optimisation problems.

Review Questions

1. Find f″(x) for f(x) = 4x⁵ − 3x³ + 7x − 2.
2. Determine the concavity of f(x) = 3x² − 12x + 1 for all x. Does the function have any inflection points?
3. Find all inflection points of f(x) = 2x³ − 9x² + 12x − 4.
4. Find the stationary points of f(x) = x³ − 3x² − 9x + 5 and classify each using the second derivative test.
5. Use the second derivative test to classify all stationary points of f(x) = x⁴ − 8x² + 3.
6. A function has f′(x) = 3(x−2)². Show that x = 2 is a stationary point but is NOT a local maximum or minimum.
7. Find the minimum value of f(x) = x² − 4x + 7 for x ≥ 0 using calculus.
8. A wire of length 24 cm is bent to form a rectangle. Find the dimensions that maximise the area.
9. A particle moves with position s(t) = 2t³ − 9t² + 12t − 3 metres for t ≥ 0. Find all local maxima and minima of the position function.
10. Find the two positive numbers whose sum is 16 and whose sum of squares is minimum.
11. Find all inflection points of f(x) = x⁴ − 6x² and state the intervals of concavity.
12. A rectangular area is to be enclosed using 80 m of fencing. Two parallel internal dividers (parallel to the width) are also included. Find the dimensions that maximise the total enclosed area.
13. For f(x) = e^x − 2x, find and classify all stationary points using the second derivative test. State the concavity for all x.
14. An open rectangular box has a square base and is to be constructed from 48 cm² of material (no lid). Find the dimensions that maximise the volume.
15. A profit function is P(x) = −x³ + 9x² − 15x − 20 (dollars, x = hundreds of items). Use all three tools — stationary points, second derivative test, and inflection point — to give a complete analysis of P.