inequality question (1 Viewer)

[whiz]

can someone pls tell me how to do this?

If a4 + b4 + c4 + d4 ≤ 4

Then prove a4 + b4 + c4 + d4 ≥ 4

(they are s'pose to be to the power of 4 or -4!)

 

[Newbie]

self-esteem -1

dont get the question sorry

 

[freaking_out]

Originally posted by Newbie
self-esteem -1

dont get the question sorry

yeah, same here, don't understand the notations.

 

[+:: $i[Q]u3 ::+]

Originally posted by whiz
can someone pls tell me how to do this?

If a4 + b4 + c4 + d4 ≤ 4

Then prove a4 + b4 + c4 + d4 ≥ 4

(they are s'pose to be to the power of 4 or -4!)

what's the last bit mean? less than/equal to four and greater than equal to four?

 

[OLDMAN]

Gather the question is :
a^4+b^4+c^4+d^4<=4 then prove
a^-4+b^-4+c^-4+d^-4>=4.

Straight application of AM-GM.

Let A=a^4, B=b^4, C=c^4 and D=d^4
then, a^-4+b^-4+c^-4+d^-4= 1/A+1/B+1/C+1/D
=(BCD+ACD+ABD+ABC)/ABCD>=(4(A^3.B^3.C^3.D^3)^.25)/ABCD AM>=GM
=4/(ABCD)^.25>=4
since (ABCD)^.25<=(A+B+C+D)/4<=1 AM>=GM

Hope this helps.

 

[ND]

Do you know whether we're allowed to quote AM-GM in the exam?

 

[OLDMAN]

Maybe not. But a question like this would have leading parts. The general proof of AM-GM for any n is hard, but do practice the "dance steps" up to n=4.

 

[fitz33]

there is a pretty easy proof for AM-GM using the function
f(x) = e^(x-1) - x

less than 10 lines anyways..