How to solve one variable given two variables and one inequality?
So I have a function:
$phi(P) = lvert (1-alpha_2)^P - (1-alpha_1)^Prvert$
It is easy to show that $phi$ is increasing between $0$ and $P_M$, where
$P_{M} = lvert lnbigg(frac{ln(1-alpha_2)}{ln(1-alpha_1)}bigg)/lnbigg(frac{1-alpha_1}{1-alpha_2}bigg)rvert$
and decreasing afterwards.
Assuming $1 > alpha_1 > alpha_2$ > 0, if $P_{M} leq 1$, how could I prove the following?
$alpha_1 > 1-frac{1}{e} > 0.63$
I have stuck here for hours. I think it is a easy step to make it. However, I couldn't figure it out.
Any idea is much appreciated. Thanks!
derivatives inequality boundary-value-problem
asked Nov 25 at 2:50
Wei
162
add a comment |
So I have a function:
$phi(P) = lvert (1-alpha_2)^P - (1-alpha_1)^Prvert$
It is easy to show that $phi$ is increasing between $0$ and $P_M$, where
$P_{M} = lvert lnbigg(frac{ln(1-alpha_2)}{ln(1-alpha_1)}bigg)/lnbigg(frac{1-alpha_1}{1-alpha_2}bigg)rvert$
and decreasing afterwards.
Assuming $1 > alpha_1 > alpha_2$ > 0, if $P_{M} leq 1$, how could I prove the following?
$alpha_1 > 1-frac{1}{e} > 0.63$
I have stuck here for hours. I think it is a easy step to make it. However, I couldn't figure it out.
Any idea is much appreciated. Thanks!
derivatives inequality boundary-value-problem
asked Nov 25 at 2:50
Wei
162
Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here.
– Shaun
Nov 25 at 4:24
add a comment |
So I have a function:
$phi(P) = lvert (1-alpha_2)^P - (1-alpha_1)^Prvert$
It is easy to show that $phi$ is increasing between $0$ and $P_M$, where
$P_{M} = lvert lnbigg(frac{ln(1-alpha_2)}{ln(1-alpha_1)}bigg)/lnbigg(frac{1-alpha_1}{1-alpha_2}bigg)rvert$
and decreasing afterwards.
Assuming $1 > alpha_1 > alpha_2$ > 0, if $P_{M} leq 1$, how could I prove the following?
$alpha_1 > 1-frac{1}{e} > 0.63$
I have stuck here for hours. I think it is a easy step to make it. However, I couldn't figure it out.
Any idea is much appreciated. Thanks!
derivatives inequality boundary-value-problem
asked Nov 25 at 2:50
Wei
162
So I have a function:
$phi(P) = lvert (1-alpha_2)^P - (1-alpha_1)^Prvert$
It is easy to show that $phi$ is increasing between $0$ and $P_M$, where
$P_{M} = lvert lnbigg(frac{ln(1-alpha_2)}{ln(1-alpha_1)}bigg)/lnbigg(frac{1-alpha_1}{1-alpha_2}bigg)rvert$
and decreasing afterwards.
Assuming $1 > alpha_1 > alpha_2$ > 0, if $P_{M} leq 1$, how could I prove the following?
$alpha_1 > 1-frac{1}{e} > 0.63$
I have stuck here for hours. I think it is a easy step to make it. However, I couldn't figure it out.
Any idea is much appreciated. Thanks!
derivatives inequality boundary-value-problem
derivatives inequality boundary-value-problem
asked Nov 25 at 2:50
Wei
162
asked Nov 25 at 2:50
Wei
162
edited Nov 25 at 20:20
asked Nov 25 at 2:50
Wei
162
asked Nov 25 at 2:50
Wei
162
asked Nov 25 at 2:50
Wei
162
162
Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here.
– Shaun
Nov 25 at 4:24
add a comment |
Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here.
– Shaun
Nov 25 at 4:24
Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here.
– Shaun
Nov 25 at 4:24
Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here.
– Shaun
Nov 25 at 4:24
add a comment |
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Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here.
– Shaun
Nov 25 at 4:24