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132 The Nuts and Bolts of Proof, Third Edition
8. We will try to construct a proof of the statement. The inequaUty
/W < g(x) is equivalent to the inequality/(x) - g{x) < 0. Therefore,
we can concentrate on proving that /(x) - g(x) < 0 for all real
numbers x > 0. Using the formulas for the functions, we obtain:
/(x) - ^(x) = x^ - x^ = x\l - x^) = x\\ - x)(l + x).
Is this product smaller than or equal to zero for all real numbers
X > 0? The product is equal to zero for x = 0 and x = 1. (We are not
considering x=: -1 because we are using only nonnegative numbers.)
What happens to the product x^(l - x)(l + x) if x is neither 0 nor 1?
The number x^ is always positive. Because x>0, l + x>l. So this
factor is always positive. Then the sign of the product is determined
by the factor 1 - x. This factor is less than or equal to zero when
X > 1. Therefore, x^(l - x)(l + x) < 0 only when x > 1. So
/(x) - ^(x) < 0 only if x > 1, and not for all x > 0. Thus, the statement
is false. Can we find a counterexample? Consider x = 0.2. Then
x^ = 0.04 and x'* = 0.0016. Therefore, in this case x^>x'*, and the
statement is false.
9. Let n be the smallest of the four counting numbers we are considering.
Then the other three numbers are n+ 1, n + 2, and n + 3. When we
add these numbers we obtain:
5 = n + (n + 1) + (n + 2) + (n + 3) = 4n + 6.
The number S is not always divisible by 4. Indeed, if /i = 1, S = 10. So,
we have found a counterexample. The sum of the four consecutive
integers 1, 2, 3, and 4 is not divisible by 4. The given statement is false.
Note: S is always divisible by 2.
10. (This statement seems to be similar to the statement: "The sum of two
odd numbers is an even number." But similarity is never a proof, and
statements that sound similar can have very different meanings. So,
we must try to construct either a proof or a counterexample.) To
prove that the function / + ^ is even we need to prove that:
if + g)(x) = (f + g)(-x)
for all real numbers. By definition of/ + g:
if + 9)ix)=fix) + gix)
{f + 9){-x)=f{-x) + g(-x).