If f and g are continuous functions defined on [a, b] so that g(x)0, then there exists a number c
[a, b] with
f(x) g(x) dx = f(c)
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Theorem 7.2.8: Mean Value Theorem for Integration |
If f and g are continuous functions defined on [a, b] so that g(x) |
m = inf{ f(x): xThen we have m
M = sup{ f(x): x
f(x) M
and since g is non-negative we also have
m g(x)By the properties of the Riemann integral this implies that
mTherefore there exists a number d between m and M such that
dBut since f is continuous on [a, b] and d is between m and M, we can apply the Intermediate Value Theorem to find a number c such that f(c) = d. Then
f(c)which is what we wanted to prove.