Calculus: Analytic Geometry and Calculus, with Vectors

5.5 The Rolle theorem and the mean-value theorem 327

S2, and Ix2 - xil < Ss. We may suppose thatxi 5 X. If xi and x2

both lie in the interval Ix - I 5 S2, then If(x2) - f(xi)I < e. If xi and

x2 both lie in theinterval a <-- x S_ - 32j then again l f(x2) - f(x')I < e.

If a C xi < - S2 and - S2 < x2 < E + S2, then

I f(x2) - f(xi) I = If(X2) - f( - S2)I + lf( - S2) - f(xi)I <2 + 2= e.

Thus, in each case, If(X2) - f (XI) I < e, so + S2 is in A and cannot be

the partition number. This contradiction shows that E= b, but to

complete the proof of the theorem, we must show that b is in the set A.

Since f has left-hand continuity at b, we can choose S' > 0 such that

a < b - S' < b and l f(x) - f(b) I < e/2 whenever b - S' S x < b.

Since b - S' must belong to A, we can choose a positive number S such

that S < S' and If(x2) - f (xi) I < e/2 whenever a <-- xl < b - S', a 5

X2 5 b - S', and lx2 - xii < S. Arguments used above show that

If(s) - f(xi) I < e whenever a G x1 5 b, a C X2 < b, and Ix2 - xi I < S.

This completes the proof of Theorem 5.58.

Problems 5.59

1 With the text out of sight, write completely accurate statements of (a)

the Rolle theorem, (b) the mean-value theorem.

(^2) Sketch several graphs that seem to be graphs of functions satisfying the
hypotheses of the Rolle theorem, and see whether it seems to be true that the
star points exist.
3 Sketch several graphs that seem to be graphs of functions which, for one
reason or another, do not satisfy the hypotheses of the Rolle theorem but never-
theless it seems to be true that star points exist anyway.
(^4) Sketch several graphs that seem to be graphs of functions which, for one
reason or another, do not satisfy the hypotheses of the Rolle theorem and star
points do not exist.
5 Tell why there is no request for construction of graphs of functions that
satisfy the hypotheses of the Rolle theorem and star points do not exist.
6 Prove that if F(x) > 0 when a < x < b, then there is at most one x
for which a < x < b and F(x) = 0. Solution: If we suppose that a < x, < x2 <
b and F(xi) = F(x2) = 0, an application of the Rolle theorem yields the con-
clusion that there is a number for which x1 < < x2 and F'() = 0. This
contradicts the hypotheses and the result follows.
7 This problem requires us to review fundamental processes of the calculus
whose validity depends upon Theorem 5.57. Supposing that f and g are func-
tions defined and continuous over an interval containing the point x = a and
that
(1) f'(x) = g(x), f(a) = A,
we can then write
(2) AX) = ta(x) dx + c,