The Mean-value Theorem

Consider the quantity Q defined by the equation

$$\frac{f(b) - f(a)}{b - a} = Q,$$ (13.1)

or

$$f(b) - f(a) - (b - a)Q = 0.$$ (13.2)

Let $F(x)$ be a function formed by replacing $b$ by $x$ in the left-hand member of (13.2); that is,

$$F(x) = f(x) - f(a) - (x - a)Q.$$ (13.3)

From (13.2), $F(b) = 0$, and from (13.3), $F(a) = 0$; therefore, by Rolle's Theorem (see §13.1), $F'(x)$ must be zero for at least one value of $x$ between $a$ and $b$, say for $x_1$. But by differentiating (13.3) we get

$$F'(x) = f'(x) - Q.$$

Therefore, since $F'(x_1) = 0$, then also $f'(x_1) - Q = 0$, and $Q = f'(x_1)$. Substituting this value of Q in (13.1), we get the Theorem of Mean Value^13.1,

$$\frac{f(b) - f(a)}{b - a} = f'(x_1),\ a < x_1 < b$$ (13.4)

where in general all we know about $x_1$ is that it lies between $a$ and $b$.

The Theorem of Mean Value interpreted Geometrically.

Let the curve in the figure be the locus of $y = f(x)$.

Figure 13.3: Geometric illustration of the Mean value theorem.

Take $OC = a$ and $OD = b$; then $f(a) = CA$ and $f(b) = DB$, giving $AE = b - a$ and $EB = f(b) - f(a)$. Therefore the slope of the chord AB is

$$\tan EAB = \frac{EB}{AE} = \frac{f(b) - f(a)}{b - a}.$$

There is at least one point on the curve between A and B (as P) where the tangent (or curve) is parallel to the chord AB. If the abscissa of P is $x_1$ the slope at P is

$$\tan\, t = f'(x_1) = \tan\, EAB.$$

Equating these last two equations, we get

$$\frac{f(b) - f(a)}{b - a} = f'(x_1),$$

which is the Theorem of Mean Value.

The student should draw curves (as the one in §13.1), to show that there may be more than one such point in the interval; and curves to illustrate, on the other hand, that the theorem may not be true if $f(x)$ becomes discontinuous for any value of $x$ between $a$ and $b$ (see Figure 13.2 (a)), or if $f'(x)$ becomes discontinuous (see Figure 13.2 (b)).

Clearing (13.4) of fractions, we may also write the theorem in the form

$$f(b) = f(a) + (b - a)f'(x_1).$$ (13.5)

Let $b = a + \Delta a$; then $b - a = \Delta a$, and since $x_1$ is a number lying between $a$ and $b$, we may write

$$x_1 = a + \theta \cdot \Delta a,$$

where $\theta$ is a positive proper fraction. Substituting in (13.4), we get another form of the Theorem of Mean Value.

$$f(a + \Delta a) - f(a) = \Delta a f'(a + \theta \cdot \Delta a), \ \ \ 0 < \theta < 1.$$ (13.6)