Suppose that a function satisfies the same conditions as in the previous exercise, except that $f(a) \neq f(b)$. One such function is illustrated in Figure 12 Show that if the figure is rotated so that the dotted line joining $A(a, f a)$ ) to $B(b, f b))$ is horizontal, then the rotated figure satisfies all the conditions of Rolle's Theorem. Therefore, there must be at least one point $c, a<c<b$ such that the tangent line at $x=c$ is parallel to the line joining $A$ to $B$. Thus, show that $f^{\prime}(c)=\frac{f(b)-f(a)}{b-a} .$ This result is known as the Mean Value Theorem, or The Law of the Mean.