SOLUTION: If a , b and c are three sides of a triangle with perimeter 1, then is bc + ca + ab less than or equal to 1/3?

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Question 446615: If a , b and c are three sides of a triangle with perimeter 1, then is bc + ca + ab less than or equal to 1/3?
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
We show that ab+%2B+bc+%2B+ca+%3C=+1%2F3.
We want to optimize ab + bc + ca subject to the constraint a + b + c = 1.
With Lagrange multiplier alpha in mind, define the function
F%28a%2Cb%2Cc%29+=+ab+%2B+bc+%2B+ca++%2B+alpha%2A%28a%2Bb%2Bc+-+1%29
Then
F%5Ba%5D+=+b%2Bc%2B+alpha+=+0,
F%5Bb%5D+=+a%2Bc%2B+alpha+=+0,
F%5Bc%5D+=+a%2Bb%2B+alpha+=+0,and
F%5Balpha%5D+=+a%2Bb%2Bc+-+1+=+0.
Solving the system of equations first for alpha, we obtain alpha+=+-2%2F3.
The system then becomes
b+c = 2/3,
a+c = 2/3, and
a+b = 2/3.
==> a+=+b+=+c+=+1%2F3
These values of a,b, and c would give optimize ab + bc + ca, we just don't know whether its a max or min. When a+=+b+=+c+=+1%2F3, ab+%2B+bc+%2B+ca+=+1%2F3.
Hence we compare the value of ab + bc + ca when a+=+1%2F4, b+=+1%2F4, and c+=+1%2F2, which gives ab+%2B+bc+%2B+ca+=+1%2F16+%2B+2%2F16+%2B+2%2F16+=+5%2F16+%3C+1%2F3. Hence a+=+b+=+c+=+1%2F3 maximizes ab + bc + ca, and
ab+%2B+bc+%2B+ca+%3C=+1%2F3.