Lesson 3.3: The Mean Value Theorem

The Mean Value Theorem (MVT) is connected but not directly related to the Intermediate Value Theorem. It’s important that you not confuse them. One is obvious at best and perhaps a little idiotic, the other is quite handy. We’ll use it shortly to prove the Fundamental Theorem of Calculus, and again in April to prove Taylor’s Theorem.

graph demonstrating the MVT

The MVT states that if we graph a function $ y=f(x)$ and draw a secant line between two points $(x_1,f(x_1))$ and $(x_2,f(x_2))$, there is some point $c$ ($ x_1 < c < x_2$) such that $\displaystyle f'(c) = \frac{f(x_2)-f(x_1)}{x_2-x_1}$ That is, there’s a point on the interval such that the tangent line at that point matches the slope of the secant line we drew.

Caveats: $ f(x)$ must be continuous on $ [a,b]$ and differentiable on $ (a,b)$. This is seldom a problem, but you certainly must know it.

graph demonstrating Rolle's Theorem

You will also hear Rolle’s Theorem spoken of. It is a special case of the Mean Value Theorem (Actually, Rolle’s was proven first, so it might be better to call the MVT a generalization of Rolle’s Theorem.) Rolle’s Theorem states that the MVT is true when $ f(a)=f(b)$, i.e., the secant line is flat, and thus so is the tangent line. My personal feeling is that that’s a pretty lame excuse for a theorem; if Rolle gets his name on a theorem, so should I.

A corollary useful in proving these theorems is one that’s probably even more obvious than the IVT: the Extreme Value Theorem, or EVT, which states that a continuous function must achieve both its maximum and its minimum value at least once each. At this point, you should smack your forehead. I mention this only because it’s a useful term to know for the sake of proof.

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