The Art and Craft of Problem Solving

y

�--------------------�----� x

9.3.28 (Putnam 2(02) Let k be a fixed positive inte­
ger. The nth derivative of I/(J!< -I) has the form
Pn(x)/(J!< _l)n+1 where Pn(x) is a polynomial. Find
Pn(I).

9.3.29 (Putnam 1998) Let f be a real function on the
real line with continuous third derivative. Prove that
there exists a point a such that

f(a).!, (a)· f"(a) · fill (a) � o.

9.3.30 (Putnam 1964) Find all continuous functions
f(x) : [0, I] -+ (0, 00) such that

10

1

f(x)dx = I

lXf(x)dx=a

l �I(x)dx = a2,

where a is a given real number.

9.3.3 1 (Bratislava 1994) Define f: [0, 1] -+ [0, I] by

f(x) = { 2x^0 :s; x :s; 1/2,

-2x+2 1/2<x:S;1.

Next, define a sequence fn of functions from [0, 1] to
[0, I] as follows: Let fl (x) = f(x) and let fn(x) =
f(tn-I (x)) for n > l. Prove that for each n,

fl fn (x)dx =�.
Jo^2
9.3.32 Compute the limit

lIm -+--+--+ ... +--.

. (I 1 1 1 )
n ..... oo n n + I n + 2 2n - 1

9.3.33 (Putnam 1995) For what pairs (a, b) of positive
real numbers does the improper integral

9.3 DIFFERENTIATION AND INTEGRATION 341

converge?

9.3.34 In Example 9.3.4, there was a constant k that

was strictly less than I, and II'(x)1 :s; k. Notice that

this is not the same as saying that If' (x) I :s; I. What

would happen in this case?

9.3.35 (Putnam 1994) Let f(x) be a positive-valued

function over the reals such that I' (x) > f(x) for all x.

For what k must there exist N such that f(x) > eU for

x>N?

9.3.36 Let f be differentiable on ( -00, 00 ) and suppose

that I' (x) =I- I for all real x. Show that f can have either

zero or one fixed point (but not more than one).

9.3.37 Compute the limit

9.3.38 (Putnam 1991) Suppose f and g are non­

constant, differentiable, real-valued functions defined

on (-00, 00 ). Furthermore, suppose that for each pair

of real numbers x and y,

f(x+ y) = f(x)f(y) - g(x)g(y),

g(x+y) = f(x)g(y)+g(x)f(y)·

If 1 '(0) = 0, prove that (f(x))^2 + (g(x))^2 = I for all x.

9.3.39 Let f: [O, I]-+lRsatisfy If(x) - f(y)l:S; (x ­

y) 2 for allx,y E [0, I]. Furthermore, suppose f(O) = O.

Find all solutions to the equation f(x) = O. Hint: as­

sume that f(x) is differentiable if you must, but this

isn't really a differentiation problem.

9.3.40 Complete the proof started in Example 9.3. 5

on page 333.

9.3.41 (Putnam 1976) Evaluate

Express your answer in the form loga -b, with a, b

both positive integers, and the logarithm to base e.

9.3.42 (Putnam 1970) Evaluate

9.3.43 (Putnam 1939) Prove that

la lxJ!' (x)dx = laJ f(a) -(t(i) + ... + f( l a J)),