Proving Question (1 Viewer)

onomatopoeia_metapho · Sep 16, 2014

Hey guys!

Would you be able to help me with this proving question?

a, b and c are numbers that are greater than or equal to zero. Prove:

2(a^3+b^3+c^3 )≥ab(a+b)+bc(b+c)+ca(c+a)

Thanks!

onomatopoeia_metapho · Sep 16, 2014

Kurosaki · Sep 16, 2014

onomatopoeia_metapho said:
Hey guys!

Would you be able to help me with this proving question?

a, b and c are numbers that are greater than or equal to zero. Prove:

2(a^3+b^3+c^3 )≥ab(a+b)+bc(b+c)+ca(c+a)

Thanks!

Consider
$a^3 + b^3=(a+b)(a^2-ab+b^2) \\ $But$ (a-b)^2 \geq 0 \implies a^2-2ab+b^2 \geq 0 \implies a^2-ab+b^2 \geq ab \\ $So therefore, $ a^3+b^3 \geq ab(a+b) \\ $Similarly then $ b^3 + c^3 \geq bc(b+c) $ and $ c^3 + a^3 \geq ca(c+a) \\ $Adding, $ 2(a^3+b^3+c^3) \geq ab(a+b) + bc(b+c) +ca(c+a)$

Proving Question (1 Viewer)

onomatopoeia_metapho

New Member

onomatopoeia_metapho

New Member

Kurosaki

True Fail Kid

Users Who Are Viewing This Thread (Users: 0, Guests: 1)