Another Advanced Calculus Question (analysis)

Let f(x) be continuous on [a,b]. Show that there exists a c in [a,b] such that f(c)=1/(b-a) times INTEGRAL(f(x)dx) (integral taken from a to b)

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Proof:

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My first thought is to use the Mean Value Theorem. BUT, the Mean Value Theorem requires that f be continuous AND differentiable. We are given continuity but not differentiability.

Any IDEAS?

Re: Another Advanced Calculus Question (analysis)

Quote:

Originally Posted by

**computerproof** Let f(x) be continuous on [a,b]. Show that there exists a c in [a,b] such that f(c)=1/(b-a) times INTEGRAL(f(x)dx) (integral taken from a to b)

Define $F(x) = \int_a^x {f(t)dt}$. Is $F$ continuous on $[a,b]$?

Now use the mean value theorem.

Re: Another Advanced Calculus Question (analysis)

F(x) must be continuous for other wise F '(x) = f(x) could not exist....is that correct?

Re: Another Advanced Calculus Question (analysis)

Quote:

Originally Posted by

**computerproof** F(x) must be continuous for other wise F '(x) = f(x) could not exist....is that correct?

Yes, $F'$ does exist and equals $f$.

Now apply the MVT to $F$ on $[a,b]$.

Re: Another Advanced Calculus Question (analysis)

Thanks.....I really appreciate it