Proof of the Mean Value Theorem

Turn the Secant Slope into a Zero Derivative

The Mean Value Theorem connects the change in a function across an interval to a derivative at an interior point. In the previous tutorial, we used the theorem and examined some of its consequences. Here we prove the theorem by reducing it to Rolle’s Theorem, which was proved earlier in the course.

The central construction is simple: subtract from the function the line joining its endpoint values. The difference is zero at both endpoints. Rolle’s Theorem then gives an interior point where the difference has derivative zero. Since the line’s slope is the original function’s secant slope, that zero-derivative condition is exactly the conclusion of the Mean Value Theorem.

The proof depends on choosing the right line. A line with the same slope as the secant through \((a,f(a))\) and \((b,f(b))\) changes by exactly as much as \(f\) between the endpoints. Subtracting that line therefore makes the adjusted function take equal values at \(a\) and \(b\).

The Endpoint-Matching Construction

Let \(m\) be the secant slope of \(f\) on \([a,b]\), and define the affine function \(\ell(x)=f(a)+m(x-a)\). This line passes through \((a,f(a))\). Because of the definition of \(m\), it also passes through \((b,f(b))\).

Proof. Substituting \(x=a\) into the definition of \(\ell\) gives \(\ell(a)=f(a)\), so \(g(a)=f(a)-\ell(a)=0\). At \(x=b\),

Thus \(g(b)=f(b)-\ell(b)=0\) as well. The function \(\ell\) is affine, hence continuous and differentiable, with \(\ell'(x)=m\). Consequently, if \(f\) is continuous on the closed interval and differentiable in its interior, the same is true of \(g=f-\ell\), and the difference rule gives \(g'(x)=f'(x)-m\). This proves the lemma. \(\square\)

The lemma separates the proof into two tasks: construct an adjusted function whose endpoint values agree, then use a theorem about functions with equal endpoint values. The regularity assumptions are preserved by the subtraction, so the adjusted function is eligible for Rolle’s Theorem whenever \(f\) is.

Proof of the Mean Value Theorem

Proof. Set

and define \(g(x)=f(x)-[f(a)+m(x-a)]\) for \(x\in[a,b]\). By the Endpoint-Matching Affine Function Lemma, \(g\) is continuous on \([a,b]\), differentiable on \((a,b)\), and satisfies \(g(a)=g(b)=0\). Rolle’s Theorem therefore gives a point \(c\in(a,b)\) such that \(g'(c)=0\). Differentiating the definition of \(g\) at this interior point yields

Hence \(f'(c)=m=(f(b)-f(a))/(b-a)\), as required. \(\square\)

This argument uses Rolle’s Theorem only after verifying every one of its hypotheses for \(g\). Continuity of \(f\) on \([a,b]\) and differentiability of \(f\) on \((a,b)\) ensure the corresponding properties of \(g\); the construction ensures the equal endpoint values. Differentiability of \(f\) at \(a\) or \(b\) is not needed.

Affine Changes Preserve the Slope Relation

The same algebra explains why adding a linear term to a function shifts both its secant slopes and its derivatives in a matching way. This is useful when a problem becomes simpler after subtracting or adding an affine function. It also confirms that the proof construction is tied to slopes, rather than to a particular choice of vertical position for the graph.

Proof. By the sum rule, the constant rule, and the derivative of the identity function,

For the secant slope, calculate directly:

Thus the two quantities for \(F\) are obtained from the corresponding quantities for \(f\) by applying the same transformation \(r\mapsto\alpha r+\beta\). If \(\alpha\ne0\), this transformation is one-to-one. Therefore the derivative and secant slope for \(F\) are equal if and only if the derivative and secant slope for \(f\) are equal. \(\square\)

Worked Proof Constructions

Why the Proof’s Details Matter

Subtracting the secant line is more than a convenient algebraic trick. It is what makes the endpoint condition required by Rolle’s Theorem hold. Subtracting a line with a different slope would generally leave unequal values at the endpoints, so Rolle’s Theorem would not apply. The slope must be exactly \((f(b)-f(a))/(b-a)\).

The roles of the hypotheses should also remain distinct. Continuity on the closed interval is needed for the Rolle argument, including at the endpoints; differentiability is needed only at interior points. The proof does not require a formula for the point \(c\), and in general it does not provide a unique point. In the worked examples we could solve the derivative equation explicitly, but for an arbitrary function the existence conclusion is still valuable even when \(c\) cannot be identified.

A reliable proof check is to verify three facts before invoking Rolle’s Theorem: the adjusted function is continuous on \([a,b]\), it is differentiable on \((a,b)\), and its endpoint values agree. After Rolle’s Theorem gives \(g'(c)=0\), compute \(g'\) from its definition and rearrange that equality. This final step is essential: the zero derivative belongs to the adjusted function, and the Mean Value Theorem conclusion concerns the derivative of the original function.