4(^3). 5
3
2.5
2
(^1). 5
(^0). 5
(^0). 5
9.3 DIFFERENTIATION AND INTEGRATION 331
y =j(x)
slope =<X
(^1). 5 2
Therefore, the graph of y = f(u) lies above the tangent line for all u 2: O. An
other way of saying this (make the substitution x = u -1) is that f(1 +x) is always
strictly greater than its linear approximation 1 + ax, except when x = 0, in which case
we have equality [corresponding to the point (1, 1) on the graph]. We have established
Bernoulli's inequality.^7 _
The Mean Value Theorem
One difficulty that many beginners have with calculus problems is confusion over what
should be rigorous and what can be assumed on faith as "intuitively obvious." This is
not an easy issue to resolve, for some of the simplest, most "obvious" statements in
volve deep, hard-to-prove properties of the real numbers and differentiable functions.^8
We are not trying to be a real analysis textbook, and will not attempt to prove all of
these statements. But we will present, with a "hand-waving" proof, one important the
oretical tool that will allow you to begin to think more rigorously about many problems
involving differentiable functions.
We begin with Rolle's theorem, which certainly falls into the "intuitively obvi
ous" category.
If f(x) is continuous on [a ,b] and differentiable on (a,b), and f(a) =
f(b), then there is a point u E (a,b) at which f'(u) = O.The "proof' is a matter of drawing a picture. There will be a local minimum or maxi
mum between a and b, at which the derivative will equal zero.
(^7) The sophisticated reader may object that we need Bernoulli's inequality (or something like it) in the first
place in order to compute I'(u) = au"-l when a is not rational. This is not true; for example, see the brilliant
treatment in [^2 9), pp.^229 -^23 1, which uses the geometry of complex numbers in a surprising way.
SA function 1 is called differentiable on the open interval (a,b) if I' (x) exists for all x E (a,b). We won't
worry about differentiability at the endpoints a and b; there is a technical problem about how limits should be
defined there.