# Two global theorems.

Theorem 6.8.1 (Rolle's Theorem)   Let be a function, continuous on and differentiable on . If , then there exists at least one point such that .

This means that, at least at one point,the graph has a tangent parallel to the axis (see Figure 6(a)).

Example 6.8.2 (see Figure 7)   Take . Then .

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Theorem 6.8.3 (Lagrange's Theorem; first form)   Let be a function, continuous on and differentiable on . There exists at least one point such that .

This means that, at least at one point,the graph has a tangent parallel to the chord (see Figure 6(b)).

Example 6.8.4   Take , with .

We have and .

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Theorem 6.8.5 (Lagrange's Theorem; second form)   Let be a function differentiable on a neighborhood of . Then for any real number , there exists such that .

Noah Dana-Picard 2007-12-28