## The Calculus Primer (2011)

### Part IX. Indeterminate Forms

### Chapter 33. THEOREM OF MEAN VALUE

**9—1.Rolle’s Theorem.** Let us consider a continuous single-valued function *y* = *f*(*x*), with zero-values at *x* = *a* and *x = b.* Let us assume that *f′*(*x*) changes continuously as *x* varies from *a* to *b*. From the figure, by Intuition, it is clear that for at least one value of *x* between *a* and *b* the tangent to the curve must be parallel to the *X*-axis, as at *P* or Q; that is, the slope of the curve is zero for at least one value between *x = a* and *x* = *b.*

This principle may be stated without formal proof as *Rolle’s Theorem,* as follows:

If *f*(*x*) = 0 when *x* = *a* and when *x* = *b*, and if both *f*(*x*) and *f′*(*x*) are continuous for all values of *x* from *x* = *a* to *x* = 6, then *f′*(*x*) will equal zero for at least one value of *x* between *a* and *b.*

What this means is that, as *x* increases from *a* to *b*, *f*(*x*) cannot *always* increase, since *f*(a) = *f*(*b*) = 0; similarly, *f*(*x*) cannot *always* decrease as *x* increases from *a* to *b.* Therefore, for at least one value of *x* between *a* and 6, *f*(*x*) must stop increasing and begin to decrease, or stop decreasing and begin to increase. We have already learned that at a turning point in a curve, the value *of f′*(*x*) is zero.

From a consideration of the following figures, it will be seen that Rolle’s Theorem does not hold if either *f*(*x*) or *f′*(*x*) is discontinuous. Thus in (A) the function is discontinuous at *x* = *c*; in (B), the first derivative is discontinuous at *x* = *c*. In both cases there is no point on the curve between *x* = *a* and *x = b* at which the tangent becomes parallel to the *X*-axis.

**9—2.Multiple Roots of an Equation.** In algebra it is shown that if *f*(*x*) is a polynomial function, and *f*(*a*) and *f*(*b*) are opposite in sign, then an odd number of roots of *f*(*x*) = 0 will lie between *a* and *b*; if *f*(a) and *f*(*b*) have the same sign, either no root, or an even number of roots, will lie between *a* and *b*.

It is also shown in algebra that if the equation *f*(*x*) = 0 has *r* roots equal to *a,* then the equation *f′*(*x*) = 0 will have (*r* − 1) roots equal to *a*. This is another way of saying that if *f*(*x*) contains a factor (*x* − *a*)* ^{r}*, then the equation

*f′*(

*x*) = 0 will have (r − 1) roots equal to

*a*; in short,

*f*(

*x*) and

*f′*(

*x*) have a common factor (

*x*−

*a*)

^{r}^{−1}. Thus an equation

*f*(

*x*) = 0 has or has not equal roots, according as

*f*(

*x*) and

*f′*(

*x)*have or do not have a common factor involving

*x.*Such equal roots are also called

*multiple roots*of an equation. To determine whether or not an equation has multiple roots, it is only necessary to determine whether

*f*(

*x*) and

*f′*(

*x*) have a common factor.

**9—3.**The **Mean Value Theorem.** The principle of Rolle’s Theorem can be formulated analytically as follows. Let the abscissas of *P*_{1} and *P*_{2} be *a* and *b*, respectively; then *RP*_{1} = *f*(*a*), *SP*_{2} = *f*(*b*), and *MP*_{2} = *f*(b) − *f*(*a*). The slope of the line *P*_{1}*P*_{2} is seen to be

Now let us denote this last fraction by the symbol *Q,* that is, let

Clearing of fractions:

*f*(*b*) − *f*(*a*) − (*b* − *a*)Q = 0.(2)

Let ø(*x*) be a function formed by replacing *b* by *x* in equation (2):

*ø*(*x*) = *f*(*x*) −(*a*) − (*x* − *a*)*Q.*(3)

From equation (2), we find ø(*b*) = 0; and from equation (3), we find *ø*(*a*) = 0. Therefore, from Rolle’s Theorem, *ø′*(*x*) must vanish for at least one value of *x* between *a* and *b;* call this value *x*_{1}. By differentiating equation (3):

*ø′*(*x*) *= f′*(*x*) − *Q.*(4)

But*ø′*(*x*_{1}) = 0;

hence*f′*(*x*1) − *Q* = 0,

or*Q* = *f′*(*x*_{1}).(5)

Substituting from equation (5) in equation (1):

where *a < x*_{1} *< b.*

Referring now to the first paragraph of §9—3, we see that there is at least one point on the curve between *P*_{1} and *P*_{2} where the tangent to the curve is parallel to the chord *P*_{1}*P*_{2}. If the abscissa of this point is called *x*_{1}*,* then the slope at that point is given by

tan *MP*_{1}*P*_{2} = tan θ = *f′*(*x*_{1}) = .

or*f*(*b*) = *f*(*a*) + (*b* − *a*)*f*–(*x*_{1}),[2]

where *x*_{1} lies between *a* and *b*. Equations [1] and [2] are alternative forms of stating the *law of the mean,* or the mean value theorem. It is a principle which has many uses in the calculus.