Application of Derivatives
Questions test rates of change, increasing and decreasing intervals, tangents and normals, maxima and minima, and absolute extrema on a closed interval. Word problems on rates and optimisation carry the most marks.
Core tools
- f is increasing on an interval where f′(x) > 0 and decreasing where f′(x) < 0.
- Critical points are where f′(x) = 0 or f′ does not exist.
- First derivative test: sign change from + to − gives a local maximum; − to + gives a local minimum.
- Second derivative test: f′(c) = 0 with f″(c) < 0 gives a maximum, f″(c) > 0 gives a minimum.
- For absolute extrema on [a, b], compare f at all critical points and at both endpoints.
- Slope of tangent at (x₀, y₀) is f′(x₀); slope of normal is −1/f′(x₀).
Worked example
For f(x) = 2x³ − 15x² + 36x + 1, f′(x) = 6x² − 30x + 36 = 6(x − 2)(x − 3). The sign pattern is positive, negative, positive, so f increases on (−∞, 2) and (3, ∞) and decreases on (2, 3). At x = 2 the derivative changes from + to −, giving a local maximum f(2) = 29; at x = 3 it changes from − to +, giving a local minimum f(3) = 28.
Traps
- Reporting the intervals of increase using f instead of f′.
- Ignoring endpoints when the question asks for absolute maximum or minimum.
- Missing critical points where f′ is undefined.
- Using the tangent slope for the normal.