
Suppose that the mean value theorem is violated. Let Lbe the set of all xin the interval from ato bsuch that y(x) <
, and likewise let Mbe the set with
y(x) >
. If the theorem is
violated, then the union of these two sets covers the entire interval from ato b. Neither one can be empty; if, for example, M was empty, then we would have
y<
everywhere and also
, but it follows directly from the definition of
the definite integral that when one function is less than another, its integral is also less than the other’s. Since y takes on values less than and greater than
, it follows from the intermediate value theorem that y takes on the
value
somewhere (intuitively, at a boundary between
L and M).
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