Applications of the Mean Value Theorem

This section applies the Mean Value Theorem to genuine physical and inequality-proving problems — the pattern exercised fully in Exercise 7.3.

Justifying a speed-limit violation. If a vehicle covers a known distance in a known time, LMVT guarantees the instantaneous speed equalled the average speed at some instant; if that average already exceeds the speed limit, the vehicle must have exceeded the limit at some point during the trip — this is exactly how "average speed over a toll stretch" speeding tickets are legally justified.

Bounding a function value from a derivative bound. If $|f'(x)|\le M$ (or $f'(x)\le M$) throughout an interval and $f$ at one endpoint is known, applying LMVT on the sub-interval to the unknown endpoint gives $f(b)-f(a)=f'(c)(b-a)\le M(b-a)$, which bounds $f(b)$ directly — without needing to know $f$ explicitly.

Proving a general inequality. Applying LMVT to $f(x)=\sin x$ between two arbitrary reals $\alpha,\beta$ gives $\dfrac{\sin\beta-\sin\alpha}{\beta-\alpha}=\cos(c)$ for some $c$ between them; since $|\cos c|\le1$ always, this yields the general inequality $|\sin\alpha-\sin\beta|\le|\alpha-\beta|$ for all real $\alpha,\beta$ — a genuinely new fact, extracted purely from the derivative bound on $\cos$, without needing any trigonometric identity. …