Differentiation and Mean Value Theorems
This domain tests the derivative as a limit, the mean value theorems and their use in comparing functions, Taylor expansion with remainder, and the precise hypotheses needed for L'Hôpital's rule. Proofs that hinge on choosing the right auxiliary function are common.
Core results
f'(c) = lim_{x→c} (f(x) − f(c))/(x − c). Differentiability at a point implies continuity there; the converse fails for |x| at 0. 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 in (a, b). The MVT gives f(b) − f(a) = f'(c)(b − a) for some interior c; Cauchy's version states (f(b) − f(a))g'(c) = (g(b) − g(a))f'(c). Consequences: a nonnegative derivative gives a nondecreasing function, and f' identically zero on an interval forces f constant there. Darboux's theorem says derivatives have the intermediate value property even when discontinuous. Taylor's formula with Lagrange remainder is f(x) = sum_{k=0}^{n} f^{(k)}(a)(x − a)^k/k! + f^{(n+1)}(ξ)(x − a)^{n+1}/(n+1)!.
Worked example
Show e^x ≥ 1 + x for all real x. Let f(x) = e^x − 1 − x, so f(0) = 0 and f'(x) = e^x − 1. For x > 0, f'(x) > 0, so f increases from f(0) = 0 and stays positive. For x < 0, f'(x) < 0, so f is decreasing as x increases toward 0, which again gives f(x) > f(0) = 0. Hence f ≥ 0 everywhere, with equality only at 0.
Traps
Applying L'Hôpital without a 0/0 or ∞/∞ form, or when the quotient of derivatives has no limit, as with (x + sin x)/x as x → ∞; using the MVT without continuity on the closed interval; assuming f' is continuous; reading Darboux as continuity of derivatives; omitting the interior-point condition in Rolle's theorem.