The Art and Craft of Problem Solving

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9.3 DIFFERENTIATION AND INTEGRATION 333

y = f( x) (fantasy)

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The picture gives us a vague idea. Since the derivative is at most k in absolute value,

and since k < 1, the graph of y = f (x) to the right of the y-axis will be trapped within

the dotted-line "cone," and will eventually intersect the graph of y = x. The mean
value theorem lets us prove this in a satisfying way. Suppose that for all x 2: 0, we
have f(x) =1= x. Then (IVT) we must have f(x) > x. Pick b > 0 (think large). By the
mean value theorem, there is a u E (O,b) such that

Since f(b) > b, we have

f'( )=

f(b)-f(O)

=

f(b)-v

u.

b-O b

b-v V

f'(u) > -

b-

= 1 -b'

Since b can be arbitrarily large, we can arrange things so that the minimum value of

f' (u) becomes arbitrarily close to 1. But this contradicts If'(u) 1 � k < 1. Thus f(x)

must equal x for some x > O.
If f(O) < 0, the argument is similar (draw the "cone" to the left of the y-axis, etc.)_

The satisfying thing about this argument was the role that the mean value theorem
played in guaranteeing exactly the right derivative values to get the desired contradic­
tion.

The next example is a rather tricky problem that uses Rolle's theorem infinitely
many times.

Example 9.3.5 (Putnam 19 92) Let f be an infinitely differentiable real-valued func­
tion defined on the real numbers. If

n =1,2,3, ... ,