The Art and Craft of Problem Solving

9.3 DIFFERENTIATION AND INTEGRATION 339

Notice that x � I /(I-x) maps 1 / 11 and 1 / 101 respectively to 11 / 10 and 101/ 100.

That's promising. Call this map 'L And miraculously (verify!), f( -r(x)) = f(x). Thus

the integral substitution x � -r(x) transforms the second integral into the third.

Fortified by this triumph, we play around more with -r and discover that it trans­

forms the first integral into the second and the third into the first!

So it's easy (with a fair bit of algebra) to write the sum of the three integrals as a

single integral. The mess has been reduced to

1

• 10
  (f (x) )^2 (1 + r(x) + q(x)) dx,


• 100
  where r(x), q(x ) are rational functions (derivatives of -r, -r 0 -r, etc.)
  Since we are in the miraculous universe of a highly contrived problem, we expect
  another miracle. Perhaps 1 + r + q is the derivative of f. That would make the inte­
  grand be f^2 df, which integrates to f^3 /3. If that doesn't happen, don't give up. Look
  for something similar. The problem is ugly, but quite instructive. You may want to
  ask, is there any "geometric" realization of this "symmetry?" _

Problems and Exercises

9.3. 11 Prove that if the polynomial P(x) and its

derivative pI (x) share the zero x = r, then x = r is zero

of multiplicity greater than I. [A zero r of P(x) has

multiplicity m if (x -r) appears m times in the fac­

torization of P( x). For example, x = I is a zero of

multiplicity 2 of the polynomial x^2 - 2x + I.J

9.3.12 Let a,b,c,d,e E lR such that

b c d e

a + - + - + - + -= o.

2 345

Show that the polynomial a + bx + cx^2 + dx'3 + ex^4 has

at least one real zero.

9.3. 13 A Fable. The following story was told by Doug

Jungreis to his calculus class at UCLA. It is not com­

pletely true.

A couple of years ago, I drove up to

the Bay Area, which is 400 miles, and I

drove fast, so it took me five hours. At

the end of the trip, I slowed down, be­

cause I didn't want to get a ticket, and

when I got off the freeway, I was trav­

eling at the speed limit. Then a police

officer pulled me over, and he said, "You

don't look like no Mario Andretti," and

then he said, "You were going a little fast

there." I said I was going the speed limit,

but he responded, "Maybe you were a

little while ago, but earlier, you were

speeding." I asked how he knew that, and

he said, "Son, by the mean value theo­

rem of calculus, at some moment in the

last five hours, you were going at exactly

80 m.p.h."

I took the ticket to court, and when

push carne to shove, the officer was un­

able to prove the mean value theorem be­

yond a reasonable doubt.

(a) Assuming that the officer could prove the mean

value theorem, would his statement have been

correct? Explain.

(b) Let us change the ending of the story so that

the officer said, "I can't prove the mean value

theorem, your Honor, but I can prove the in­

termediate value theorem, and using this, I can

show that there was a time interval of exactly

one minute during which the defendant drove at

an average speed of 80 miles per hour." Explain

his reasoning.

9.3. 14 Finish up Example 9.3.7 by discussing the x <

o case.

9.3. 15 (Putnam 1994) Find all c such that the graph of

the function x^4 + 9x'3 + cx^2 + ax + b meets some line in