Two global theorems.

Theorem 6.8.1 (Rolle's Theorem)   Let $f$ be a function, continuous on $[a,b]$ and differentiable on $(a,b)$ . If $f(a)=f(b)$ , then there exists at least one point $c \in (a,b)$ such that $f'(c)=0$ .

Figure 6: Global Theorems.

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

Example 6.8.2 (see Figure 7)   Take $f(x)=\sqrt{1-x^2}$ . Then $f'(x)=\frac {-2x}{2\sqrt{1-x^2}}=\frac {-x}{\sqrt{1-x^2}}$ .

$f'(x)=0 \Longleftrightarrow x=0$ .

Figure 7: Rolle's Theorem.

Theorem 6.8.3 (Lagrange's Theorem; first form)   Let $f$ be a function, continuous on $[a,b]$ and differentiable on $(a,b)$ . There exists at least one point $c \in (a,b)$ such that $f'(c)= \frac {f(b)-f(a)}{b-a}$ .

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

Example 6.8.4   Take $f(x)=x^2$ , with $x \in [0,2]$ .

We have $f'(x)=2x$ and $\frac {f(2)-f(0)}{2-0}=\frac 42 = 2$ .

$f'(x)=2 \Longleftrightarrow 2x=2 \Longleftrightarrow x=1$ .

Theorem 6.8.5 (Lagrange's Theorem; second form)   Let $f$ be a function differentiable on a neighborhood of $x_0$ . Then for any real number $h$ , there exists $\theta \in (0,1)$ such that $f(x)= f(x_0)+f'(x_0+\theta h)$ .