When is it true that $(y_1-x_1)^p + (y_2 -x_2)^p leq (y_2 - x_1)^p + (y_1 - x_2)^p$?

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Consider four integers $x_1, x_2, y_1, y_2$ with $x_2 > x_1$ and $y_2 > y_1$. Is it true that the only integer values of $p$ so that:



$$(y_1-x_1)^p + (y_2 -x_2)^p leq (y_2 - x_1)^p + (y_1 - x_2)^p$$



are $p=0, 1$ and all even $p geq 2$?




algebra-precalculus inequality




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  • how do you define $(-1)^1.5$?
    – Calvin Khor
    Jul 22 at 9:56











  • @CalvinKhor Thanks. I have fixed the question.
    – Anush
    Jul 22 at 10:47










  • It’s true for $p=2$ as well.
    – Michael Hoppe
    Jul 22 at 16:41










  • @MichaelHoppe Thank you. Is it true for any other integers $p$?
    – Anush
    Jul 22 at 16:42










  • Just play a around with some intergers and make a guess.
    – Michael Hoppe
    Jul 22 at 16:44














up vote
0
down vote

favorite












Consider four integers $x_1, x_2, y_1, y_2$ with $x_2 > x_1$ and $y_2 > y_1$. Is it true that the only integer values of $p$ so that:



$$(y_1-x_1)^p + (y_2 -x_2)^p leq (y_2 - x_1)^p + (y_1 - x_2)^p$$



are $p=0, 1$ and all even $p geq 2$?




algebra-precalculus inequality




share|cite|improve this question





















  • how do you define $(-1)^1.5$?
    – Calvin Khor
    Jul 22 at 9:56











  • @CalvinKhor Thanks. I have fixed the question.
    – Anush
    Jul 22 at 10:47










  • It’s true for $p=2$ as well.
    – Michael Hoppe
    Jul 22 at 16:41










  • @MichaelHoppe Thank you. Is it true for any other integers $p$?
    – Anush
    Jul 22 at 16:42










  • Just play a around with some intergers and make a guess.
    – Michael Hoppe
    Jul 22 at 16:44












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Consider four integers $x_1, x_2, y_1, y_2$ with $x_2 > x_1$ and $y_2 > y_1$. Is it true that the only integer values of $p$ so that:



$$(y_1-x_1)^p + (y_2 -x_2)^p leq (y_2 - x_1)^p + (y_1 - x_2)^p$$



are $p=0, 1$ and all even $p geq 2$?




algebra-precalculus inequality




share|cite|improve this question













Consider four integers $x_1, x_2, y_1, y_2$ with $x_2 > x_1$ and $y_2 > y_1$. Is it true that the only integer values of $p$ so that:



$$(y_1-x_1)^p + (y_2 -x_2)^p leq (y_2 - x_1)^p + (y_1 - x_2)^p$$



are $p=0, 1$ and all even $p geq 2$?





algebra-precalculus inequality





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 22 at 16:57
























asked Jul 22 at 9:44









Anush

190224




190224











  • how do you define $(-1)^1.5$?
    – Calvin Khor
    Jul 22 at 9:56











  • @CalvinKhor Thanks. I have fixed the question.
    – Anush
    Jul 22 at 10:47










  • It’s true for $p=2$ as well.
    – Michael Hoppe
    Jul 22 at 16:41










  • @MichaelHoppe Thank you. Is it true for any other integers $p$?
    – Anush
    Jul 22 at 16:42










  • Just play a around with some intergers and make a guess.
    – Michael Hoppe
    Jul 22 at 16:44
















  • how do you define $(-1)^1.5$?
    – Calvin Khor
    Jul 22 at 9:56











  • @CalvinKhor Thanks. I have fixed the question.
    – Anush
    Jul 22 at 10:47










  • It’s true for $p=2$ as well.
    – Michael Hoppe
    Jul 22 at 16:41










  • @MichaelHoppe Thank you. Is it true for any other integers $p$?
    – Anush
    Jul 22 at 16:42










  • Just play a around with some intergers and make a guess.
    – Michael Hoppe
    Jul 22 at 16:44















how do you define $(-1)^1.5$?
– Calvin Khor
Jul 22 at 9:56





how do you define $(-1)^1.5$?
– Calvin Khor
Jul 22 at 9:56













@CalvinKhor Thanks. I have fixed the question.
– Anush
Jul 22 at 10:47




@CalvinKhor Thanks. I have fixed the question.
– Anush
Jul 22 at 10:47












It’s true for $p=2$ as well.
– Michael Hoppe
Jul 22 at 16:41




It’s true for $p=2$ as well.
– Michael Hoppe
Jul 22 at 16:41












@MichaelHoppe Thank you. Is it true for any other integers $p$?
– Anush
Jul 22 at 16:42




@MichaelHoppe Thank you. Is it true for any other integers $p$?
– Anush
Jul 22 at 16:42












Just play a around with some intergers and make a guess.
– Michael Hoppe
Jul 22 at 16:44




Just play a around with some intergers and make a guess.
– Michael Hoppe
Jul 22 at 16:44















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