- #1
kpoltorak
- 15
- 0
Homework Statement
Show that [tex]\forall a,b \in R[/tex]:
[tex]\left|ab\right|\leq\frac{1}{2}(a^{2}+b^{2})[/tex]
Homework Equations
Triangle Inequality seems to be useless.
The Attempt at a Solution
[tex](a+b)^{2}=a^{2}+b^{2}+2ab[/tex]
[tex]2ab=(a+b)^{2}-(a^{2}+b^{2})[/tex]
[tex]ab=\frac{1}{2}(a+b)^{2}-\frac{1}{2}(a^{2}+b^{2})[/tex]
[tex]\left|ab\right|=\left|\frac{1}{2}(a+b)^{2}-\frac{1}{2}(a^{2}+b^{2})\right|[/tex]
[tex]\left|ab\right|=\left|\frac{1}{2}(a^{2}+b^{2})-\frac{1}{2}(a+b)^{2}\right|[/tex]