Both theorems below guarantee the existence of a point where the tangent behaves in a special way — they do not by themselves tell you how many such points there are, or give a formula beyond the equation used to solve for them.
Intermediate Value Theorem. If f is continuous on [a,b], then f takes every value between f(a) and f(b) somewhere in [a,b].
Rolle's Theorem. If f(x) is continuous on the closed interval [a,b], differentiable on the open interval (a,b), and f(a)=f(b), then there is at least one c∈(a,b) with f′(c)=0.
Geometric meaning: if the endpoint heights match, somewhere in between the tangent must be horizontal (parallel to the x-axis).
Rolle's theorem can also be used indirectly, without solving an equation, to bound how many real roots an equation can have in an interval: if f had two roots α<β in (a,b) with f continuous/differentiable there, Rolle's theorem would force a zero of f′ strictly between them — so if f′(x)=0 throughout (a,b), f can have at most one root there.
Failure modes for Rolle's theorem (why it does not apply): continuity fails on [a,b] (e.g. an undefined or infinite point inside the interval), differentiability fails somewhere in (a,b) (e.g. a corner, as in ∣x∣-type functions), or simply f(a)=f(b).
Lagrange's Mean Value Theorem (LMVT). If f(x) is continuous on [a,b] and differentiable on (a,b) (with f(a),f(b) not necessarily equal), then there is at least one c∈(a,b) with
f′(c)=b−af(b)−f(a).
Rolle's theorem is the special case f(a)=f(b) (LMVT with right side =0) — it is sometimes called the "rotated Rolle's theorem."
Geometric meaning: the tangent at some interior point is parallel to the secant joining the two endpoints — equivalently, the instantaneous rate of change equals the average rate of change over [a,b] at some interior instant.
Consequences of LMVT (used throughout the rest of the chapter). …