MATH 1151 Question (1 Viewer)

[Shadowless]

a) By expanding ' ( x - y ) ^ 2 ' prove that ' (x ^ 2 ) + ( y ^ 2 ) ' is GREATER THAN or EQUAL to ' 2 x y ' for all real numbers x, y.

b) Deduce that ' (a + b ) / 2 ' is GREATER THAN or EQUAL to ' sqrt (ab) ' for all non-negative real numbers a, b. When does equality hold?

Is the answer: "When 'x' and 'y' are equal." or... am i misinterpreting this question?

 

[SpiralFlex]

For the next part, "deduce", we can substitute

Equality holds iff a=b

 

[Shadowless]

Noo... i did that and i deduced the answer. It's just the part when it asks "When does equality hold?" that I'm confused about.

 

[kaz1]

first year noob there is a maths help thingo level 4 of the red centre

 

[SpiralFlex]

Noo... i did that and i deduced the answer. It's just the part when it asks "When does equality hold?" that I'm confused about.

Consider what we have proved.



When does the side have equality to the other side? When a=b. Check it to see. Though what Kaz said is partially right (in saying a=0, b=0 the equality will hold, it is really a subset of what I have said.)

 

[Shadowless]

I kinda said that too... didn't I? LMAO but thanks for verifying.

 

[SpiralFlex]

I kinda said that too... didn't I? LMAO but thanks for verifying.

Yes, I know you said it. You asked for verification.

 

[Shadowless]

c) Use the result above to find the minimum value of ' y = (x^2) + 1 / (x^2) '.

so from a) i could just write...

from (a) : (x^2) + 1 / (x^2) is GREATER THAN or EQUAL to 2

and from that could I say the 'minimum value of blah blah' will be 2? or is there something more i need to say?

 

[SpiralFlex]

c) Use the result above to find the minimum value of ' y = (x^2) + 1 / (x^2) '.

so from a) i could just write...

from (a) : (x^2) + 1 / (x^2) is GREATER THAN or EQUAL to 2

and from that could I say the 'minimum value of blah blah' will be 2? or is there something more i need to say?

Say when the equality holds. Can you see it? There are two possible ways this equality can hold.

 

[Shadowless]

OHH... took me a while to understand what you said... So... basically after I state 'the minimum value of blah blah will be 2' I would also say for what values of 'x' will yield the minimum value?

i.e. The minimum value of ' y = (x^2) + 1/(x^2) ' will be 2 and this value will be obtained when 'x' is 1 or -1.

Right?

 

[SpiralFlex]

OHH... took me a while to understand what you said... So... basically after I state 'the minimum value of blah blah will be 2' I would also say for what values of 'x' will yield the minimum value?

i.e. The minimum value of ' y = (x^2) + 1/(x^2) ' will be 2 and this value will be obtained when 'x' is 1 or -1.

Right?

Yes very well done.

 

[Shadowless]

Yes very well done.

Not sure if you're mocking me or...

But thanks for your help. XD

 

[SpiralFlex]

Not sure if you're mocking me or...

But thanks for your help. XD

No problem. Was not mocking you, was encouraging and praising you.

 

[Shadowless]

Okay... what I don't understand is why is the answer '12978189' as opposed to '12978188'. Shouldn't you be rounding down? =/

 

[Shadowless]

BUMP

 

[unsw-maths-guru]

View attachment 27807

Okay... what I don't understand is why is the answer '12978189' as opposed to '12978188'. Shouldn't you be rounding down? =/

If the calculation had ended up as x ~ 1.5 that would mean that the number was 10^1.5 which is between 10 and 10^2, and so has two digits.

 

[Shadowless]

Hmm... Makes sense.

But would I have to provide an explanation? Or could I just write:

Therefore there would be '12978189' digits in the decimal expansion.

?