HARDER inequalities HELP needed (1 Viewer)

[blackops23]

Hi guys, couple of questions i can't do with "considering the difference"

Q1. Given that a+b=1, show 1/a + 1/b >= 4, for a>0, b>0

Q2 Prove that for positive no's a, b, c, d
(i) (a+b)(b+c)(c+a)>=8abc
(ii) (a+b)(b+c)(c+d)(d+a)>=16abcd

Thanks appreciate the help

 

[deterministic]

Hi guys, couple of questions i can't do with "considering the difference"

Q1. Given that a+b=1, show 1/a + 1/b >= 4, for a>0, b>0

Q2 Prove that for positive no's a, b, c, d
(i) (a+b)(b+c)(c+a)>=8abc
(ii) (a+b)(b+c)(c+d)(d+a)>=16abcd

Thanks appreciate the help

For both questions, I shall use the fact:
for x,y>0
(you should know how to prove this.

Q1) Noting a+b=1

 

[deterministic]

Since holds for any x,y>=0, we can substitute x with sqrt(x), y with sqrt(y) to get:


Now I have shown you this, Q2 should be very easy to solve using the above. Have a go yourself

 

[blackops23]

For both questions, I shall use the fact:
for x,y>0
(you should know how to prove this.

Q1) Noting a+b=1

I'm sorry but how did you get from (a+b)/ab to (a+b)^2/ab??

 

[blackops23]

Since holds for any x,y>=0, we can substitute x with sqrt(x), y with sqrt(y) to get:


Now I have shown you this, Q2 should be very easy to solve using the above. Have a go yourself

Thanks for this, just check my solution please:

so (a+b)>=2root(ab)
(b+c)>=2root(bc)
(c+a)>=2root(ac)

Now i don't know about inequality rules or what, but are you allowed to multiply everything on either side, because that is the only way i see to get the answer

thanks

 

[deterministic]

I'm sorry but how did you get from (a+b)/ab to (a+b)^2/ab??

(a+b)=1 so (a+b)^2=(a+b)

 

[deterministic]

Thanks for this, just check my solution please:

Now i don't know about inequality rules or what, but are you allowed to multiply everything on either side, because that is the only way i see to get the answer

thanks

you are allowed to multiply everything on either side provided both sides are non negative . Think about it intuitively, a large number multiplied by a large number will be bigger than a small number multiplied by a small number.

 

[blackops23]

you are allowed to multiply everything on either side provided both sides are non negative . Think about it intuitively, a large number multiplied by a large number will be bigger than a small number multiplied by a small number.

thanks,

couple more questions don't know how to do, like this one:

x>0, y>0, z>0 x+y = s
prove that:

1/x^2 + 1/y^2 = 8/s^2

so a solution with explanation would be immensely helpful

thanks

 

[xV1P3R]

x = 1, y = 1, s = 2
LHS = 2
RHS = 4
LHS =/= RHS !!!

 

[blackops23]

x = 1, y = 1, s = 2
LHS = 2
RHS = 4
LHS =/= RHS !!!

not sure it works that way...

 

[xV1P3R]

Wow, I noobed it hard. The one I chose was one of the few that worked (must be lack of sleep).

Let's try again

x = 1, y = 2, s = 3 => 1.25 =/=0.8888
x = 2, y = 3, s = 5 => 0.3611 =/= 0.32

etc.

I'm thinking it only works if x = y
s = 2x
LHS = 1/x² + 1/x² = 2/x²
RHS = 8/(2x)² = 8/4x² = 2/x²

Are you sure your question isn't an inequality?