Let be a continous function and . If be differentiable over , prove that there exist such that .

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- Jun 5th 2010, 08:52 PMbigliAn Inequality ?
Let be a continous function and . If be differentiable over , prove that there exist such that .

- Jun 5th 2010, 09:06 PMchoovuck
- Jun 5th 2010, 09:16 PMrebghb
Try the mean value inequality

- Jun 6th 2010, 02:54 AMbigli
Yes, rebghb was right and I fixed it.NOW, How does I solve it?

- Jun 7th 2010, 05:43 AMbigli
Please HINT me.

- Jun 8th 2010, 04:04 AMchoovuck
- Jun 8th 2010, 04:27 AMchoovuck
ah, ok, i think f(0)=f(1/2)=0 is the right condition.

i am assuming that f is actually a positive on [0,1] function, since if not then we can essentially take |f| instead of f.

denote . assume for any ,

first consider interval . by the mean value theorem there: (since f(0)=0), where . therefore for . therefore .

same arguments for (using mean value theorem at the point 1/2, where f is equal to zero) show .

finally consider . again, by the mean value theorem: (since f(1/2)=0), where . therefore for . therefore .

combining, we get , contradiction.