Integral of a function and its inverse - Real analysis

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alexmufc1995
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The following questing concerns a strictly increasing function f on [0,a] st f(0)=0

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I think I've done the first part, the second I could do with some help please. I'm pretty sure there's a typo in the hint (first inequality on the final line).Thanks
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0x2a
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Yea, pretty sure it's a typo as well. If it isn't the typo then the last inequality should be \displaystyle \int^{\alpha}_{f^{-1}(\beta)} f(x) dx \geq \beta(\alpha - f^{-1}(\beta)) I believe.
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atsruser
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(Original post by alexmufc1995)
The following questing concerns a strictly increasing function f on [0,a] st f(0)=0

Name:  Screen Shot 2015-05-19 at 18.39.57.png
Views: 179
Size:  45.9 KB

I think I've done the first part, the second I could do with some help please. I'm pretty sure there's a typo in the hint (first inequality on the final line).Thanks
I guess they're inviting you to consider the integrals \int_0^\alpha + \int_0^{f(\alpha)} + \int_{f(\alpha)}^\beta if I've read the inequalities properly.
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alexmufc1995
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(Original post by atsruser)
I guess they're inviting you to consider the integrals \int_0^\alpha + \int_0^{f(\alpha)} + \int_{f(\alpha)}^\beta if I've read the inequalities properly.
I agree, but I still can't work out how to use the hint to get the required inequality
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atsruser
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(Original post by alexmufc1995)
I agree, but I still can't work out how to use the hint to get the required inequality
Doesn't it follow trivially from part (a) + the inequality that they suggest?
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alexmufc1995
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(Original post by atsruser)
Doesn't it follow trivially from part (a) + the inequality that they suggest?
But I need to show the inequality they suggest? That's really what I don't understand how to do.

That is, I know how to complete the question once I've established the hint inequality holds.
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atsruser
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(Original post by alexmufc1995)
But I need to show the inequality they suggest? That's really what I don't understand how to do.

That is, I know how to complete the question once I've established the hint inequality holds.
It's a bit late for this kind of stuff, but I think:

1. Show that \int_0^{f(a)} f^{-1}(x) dx \le af(a)
2. \int_{f(\alpha)}^\beta = \int_{f(\alpha)}^0 + \int_0^\beta = \int_0^\beta - \int_0^{f(\alpha)}

or something along those lines.
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