Differential calculus theorems
Theorems about derivatives, extreme values and rates of change.
Rolle's theorem
f'(c) = 0 \text{ for some } c \in (a, b)
Rolle's theorem: if f is continuous on [a, b], differentiable on (a, b) and f(a) = f(b), then f'(c) = 0 for some c. Graph, counterexamples, examples.
Mean value theorem
f'(c) = \frac{f(b) - f(a)}{b - a}
The mean value theorem: a smooth function on [a, b] has a point where the tangent slope equals the average slope. Statement, graph, counterexamples, examples.
Fermat's theorem
f'(c) = 0
Fermat's theorem: if f has a local max or min at c and f'(c) exists, then f'(c) = 0. Statement, graph, why the converse fails, and worked examples.
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