Solution:
- First, by the extreme value theorem the function f assumes its
maximum value m at some x1. (That is,
f(x1) = m which is at least as great as
f(x) for any other value of x.)
- Define a function g: [0,1] ----> R by the rule
g(x) = x + f(x).
Then
g(0) = 0 and g(1) = 1.
- So, by the Intermediate Value Theorem there is some x2
between 0 and 1 so that
g(x2) = x1.
- Now define a function h by the rule
h(x) =
f(x) - f(g(x)).
(To assure that this definition makes sense we will agree that
f(x) = 0 for all values of x outside of the interval
[0,1].)
- Now
h(x1) =
f(x1) - f(g(x1))
= m - f(g(x1))
which is positive or zero (because m is the maximum value
for the function f).
- But also,
h(x2) =
f(x2) - f(g(x2))
= f(x2) - f(x1) =
f(x2) - m,
which is negative or zero.
- By the Intermediate Value Theorem there is a value
x0 between x1 and
x2 so that h(x0) = 0, which
is the same as
f(x0) =
f(x0 + f(x0)).
- So the points x=x0 and
x=x0+f(x0) on the
x-axis are the base corners of an inscribed square.