Using the Mean Value Theorem

Think of a road trip. If you average 60 mph over an hour, you didn't necessarily drive 60 mph the whole time, but your speedometer must have read exactly 60 at some instant. You can't average 60 by staying under the whole way and never hitting it, and you can't jump from under 60 to over 60 without crossing it, as long as your speed changes continuously. That is the Mean Value Theorem: if f is continuous on [a,b] and differentiable on (a,b), there is some c in (a,b) where f'(c) equals the average rate of change, (f(b)-f(a))/(b-a).

The proof is a picture more than an equation. Draw the secant line connecting (a, f(a)) and (b, f(b)); its slope is the average rate of change. Now imagine sliding a parallel line in from outside until it just touches the curve, tangent at some point c. Because the curve is smooth with no breaks or corners, that tangent line must exist somewhere between a and b, and its slope, f'(c), matches the secant's slope exactly. Both hypotheses matter: continuity rules out jumps, differentiability rules out corners, and either failure can make a guaranteed c vanish.

This theorem is the hinge between local and global information in the unit: it lets you turn a single derivative value into a statement about the function's behavior over a whole stretch, which is exactly the logic behind proving where functions increase, decrease, or stay flat. Later, when you meet the Mean Value Theorem for integrals, the same idea reappears in reverse, guaranteeing a point where a function's actual value matches its average value over an interval, rather than a slope matching an average slope.