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Im having trouble finding a certain generalization of the mean val

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I\'m having trouble finding a certain generalization of the
mean value theorem for integrals. I think my conjecture is
true, but I haven\'t been able to prove it - so maybe it
isn\'t.
Is the following true?
If F:U?Rn+1?W?Rnis a continuous function
and x:I?R?V?Rnis a continuous function
then ?t??[t1,t2]such that
?t2t1F(x(t),t)dt=F(x(t?),t?)(t2?t1)
I can see that it holds for each of the component functions
of F, but I\'m not sure about the whole thing.
Solution
g(t)=F(x(t),t)is a function from [t1,t2]?Rto Rn, so by
definition one integrates it component by component with
respect to the standard basis.
For each component gk, there exists t?k?[t1,t2]such that
(t2?t1)gk(t?k)=?t2t1gk(t)dtby the mean value theorem

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I\'m having trouble finding a certain generalization of the mean value theorem for integrals. I think my conjecture is true, but I haven\'t been able to prove it - so maybe it isn\'t. Is the following true? If F:U?Rn+1?W?Rnis a continuous function and x:I?R?V?Rnis a continuous function then ?t??[t1,t2]such that ?t2t1F(x(t),t)dt=F(x(t?),t?)(t2?t1) I can see that it holds for each of the component functions of F, but I\'m not sure about the whole thing. Solution g(t)=F(x(t),t)is a function from [t1,t2]?Rto Rn, so by definition one integrates it component by component with respect to the stand ...
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Anonymous
Really useful study material!

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