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How can I prove that the following function is increasing according to x1?

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0

$begingroup$

Suppose that
$0 le {X_1} < {X_2} < {X_3}$
.
How is it possible to prove the following function is increasing based on
${X_1}$
in the range of
$0 le {X_1} < {X_2}$ ?

$f({X_1},{X_2},{X_3}) = frac{{(c + p{X_1}){{({X_2} - {X_3})}^2} + (c + p{X_2}){{({X_1} - {X_3})}^2} + (c + p{X_3}){{({X_1} - {X_2})}^2}}}{{{{[{X_1}{X_2}{mkern 1mu} log frac{{{X_1}}}{{{X_2}}} + {X_1}{X_3}log frac{{{X_3}}}{{{X_1}}} + {X_2}{X_3}log frac{{{X_2}}}{{{X_3}}}]}^2}}}$

$c$ and $p$ are positive constants.
I have plotted this function (as a function of ${X_1}$) and I observed that this is an increasing function. Then I tried to prove it analytically but taking derivative makes it too complicated, because I would come up with lots of conditions.

I also considered the numerator and denominator separately to investigate the behavior of each function, but it did not work. Because there was a contradiction in the common part of the conditions (ranges in which the derivative of numerator is positive and derivative of denominator is negative).

The most important point is that I should not put any constraint on ${X_1}$, but any condition is allowed to put on the ${X_2}$ and ${X_3}$.

Any kind of help would be so appreciated!

derivatives logarithms

share|cite|improve this question

asked Mar 14 at 11:39

HamedHamed

62

$endgroup$

add a comment | 

0

$begingroup$

Suppose that
$0 le {X_1} < {X_2} < {X_3}$
.
How is it possible to prove the following function is increasing based on
${X_1}$
in the range of
$0 le {X_1} < {X_2}$ ?

$f({X_1},{X_2},{X_3}) = frac{{(c + p{X_1}){{({X_2} - {X_3})}^2} + (c + p{X_2}){{({X_1} - {X_3})}^2} + (c + p{X_3}){{({X_1} - {X_2})}^2}}}{{{{[{X_1}{X_2}{mkern 1mu} log frac{{{X_1}}}{{{X_2}}} + {X_1}{X_3}log frac{{{X_3}}}{{{X_1}}} + {X_2}{X_3}log frac{{{X_2}}}{{{X_3}}}]}^2}}}$

$c$ and $p$ are positive constants.
I have plotted this function (as a function of ${X_1}$) and I observed that this is an increasing function. Then I tried to prove it analytically but taking derivative makes it too complicated, because I would come up with lots of conditions.

I also considered the numerator and denominator separately to investigate the behavior of each function, but it did not work. Because there was a contradiction in the common part of the conditions (ranges in which the derivative of numerator is positive and derivative of denominator is negative).

The most important point is that I should not put any constraint on ${X_1}$, but any condition is allowed to put on the ${X_2}$ and ${X_3}$.

Any kind of help would be so appreciated!

derivatives logarithms

share|cite|improve this question

asked Mar 14 at 11:39

HamedHamed

62

$endgroup$

add a comment | 

$begingroup$

Suppose that
$0 le {X_1} < {X_2} < {X_3}$
.
How is it possible to prove the following function is increasing based on
${X_1}$
in the range of
$0 le {X_1} < {X_2}$ ?

$f({X_1},{X_2},{X_3}) = frac{{(c + p{X_1}){{({X_2} - {X_3})}^2} + (c + p{X_2}){{({X_1} - {X_3})}^2} + (c + p{X_3}){{({X_1} - {X_2})}^2}}}{{{{[{X_1}{X_2}{mkern 1mu} log frac{{{X_1}}}{{{X_2}}} + {X_1}{X_3}log frac{{{X_3}}}{{{X_1}}} + {X_2}{X_3}log frac{{{X_2}}}{{{X_3}}}]}^2}}}$

$c$ and $p$ are positive constants.
I have plotted this function (as a function of ${X_1}$) and I observed that this is an increasing function. Then I tried to prove it analytically but taking derivative makes it too complicated, because I would come up with lots of conditions.

I also considered the numerator and denominator separately to investigate the behavior of each function, but it did not work. Because there was a contradiction in the common part of the conditions (ranges in which the derivative of numerator is positive and derivative of denominator is negative).

The most important point is that I should not put any constraint on ${X_1}$, but any condition is allowed to put on the ${X_2}$ and ${X_3}$.

Any kind of help would be so appreciated!

derivatives logarithms

share|cite|improve this question

asked Mar 14 at 11:39

HamedHamed

62

$endgroup$

share|cite|improve this question

asked Mar 14 at 11:39

HamedHamed

62

share|cite|improve this question

asked Mar 14 at 11:39

HamedHamed

62

asked Mar 14 at 11:39

HamedHamed

62

asked Mar 14 at 11:39

HamedHamed

62