[Harder 3-Unit] Inequalities Question (1 Viewer)

[namelyanonymous]

I can't seem to get this one:

Show that a^2 + b^2 + c^2 >= ab + bc + ca and hence show that:

b)

I got the first part, its just (b) that I dont know how to work with :/ any hints greatly appreciated.

 

[Carrotsticks]

From the first identity, multiply both sides by (a+b+c), expand and simplify.

Then replace a^3 with a^2, b^3 with b^2 and c^3 with c^2 (or x^2,y^2,z^2 respectively if you please)

 

[namelyanonymous]

From the first identity, multiply both sides by (a+b+c), expand and simplify.

Then replace a^3 with a^2, b^3 with b^2 and c^3 with c^2 (or x^2,y^2,z^2 respectively if you please)

Thankyou thankyou thankyou it worked out! What made you think of that method?

 

[Carrotsticks]

Thankyou thankyou thankyou it worked out! What made you think of that method?

The identity we had to prove is essentially the AM/GM inequality, which in its standard form has a^3, b^3 and c^3.

The expression we had so far as a^2, b^2 and c^2, so the most logical way of acquiring a^3, b^3 and c^3 was to multiply both sides by (a+b+c), and also noticing that by doing so, a whole lot of terms will cancel.