The mean value theorem is a surprising Calculus result that states for any function fff differentiable on (a,b)(a,b)(a,b)1 there exists x∈(a,b)x(a,b)x∈(a,b) such that f′(x)=f(b)−f(a)b−af′(x)=b−af(b)−f(a) Here are three informal intuitions for what this means (all of them say the same thing in different ways): Physical example. If you travel 60 miles in one hour, at some point you must have been traveling exactly 60 miles per hour.2 Geometric intuition. There exists a line t...
