Continuity, Differentiability and Applications of Derivatives
What is tested
This domain covers the three-part continuity test at a point, the relationship between continuity and differentiability, Rolle's and Lagrange's Mean Value Theorems, and applications to rates of change, monotonicity and maxima–minima.
Core rules
- f is continuous at a when the left limit, right limit and f(a) all agree.
- Differentiable implies continuous; continuous does not imply differentiable (|x| at 0). For piecewise functions, check left and right derivatives separately.
- On an interval, f is increasing where f′(x) > 0 and decreasing where f′(x) < 0.
- Second derivative test: f′(c) = 0 with f″(c) < 0 gives a local maximum; f″(c) > 0 gives a local minimum.
- On [a,b], compare the value of f at critical points with f(a) and f(b) for absolute extrema.
Worked example
f(x) = x³ − 3x² + 4. f′(x) = 3x² − 6x = 3x(x − 2), so critical points are x = 0 and x = 2. Since f″(x) = 6x − 6, f″(0) = −6 < 0, giving a local maximum f(0) = 4; f″(2) = 6 > 0, giving a local minimum f(2) = 8 − 12 + 4 = 0. On [−1, 3], f(−1) = 0 and f(3) = 4, so the absolute maximum is 4 and the absolute minimum is 0.
Marks-losing traps
Checking only f(a) for continuity; using the second derivative test when f″(c) = 0 instead of falling back to the first derivative; omitting endpoints in an absolute-extremum question; applying Rolle's theorem without verifying continuity on [a,b] and differentiability on (a,b).