Question

If a+b+c=1 , find maximum value of (ab+bc+ac).
a,b,c are positive numbers.

Answer 1

what is find maxm

Answer 2

in my point of view. I would say a=0.33 b=0.33 and c=0.34

Answer 3

then why a=0.1,b=0.4,c=0.5 is incorrect?

Answer 4

Now i shall speak from my non-existent knowledge. Whatever number makes 1 is common sense. so it could be a=0.9 b=0.05 c=0.05

Answer 5

but is you said MAX value. I would re-correct myself and say a= 0.3 b= 0.3 c=0.4

Answer 6

see ques. carefully , i want maximum value of ab+bc+ac.

Answer 7

try squaring both sides, there you can find ab+bc+ac

Answer 8

A Mathematica solution is attached.

Answer 9

@Luis_Rivera, you are wrong

Answer 10

the maximum value you can achieve here only when a, b and c are equal. That's 3a=1, so each equals 1/3. Therefore, by substiting to equation that's given you will get 1/3. So, 1/3 is the maximum value

Answer 11

using Lagrangian setup, you get the three conditions as:
\[
b+c=\lambda\\
a+c=\lambda\\
b+a=\lambda\\
a+b+c=1
\]
solving these four equations, we see that \[
b-a=0\implies a=b\\
c-a=0\implies a=c\\
a=b=c={\lambda\over2}\\
{3\lambda\over2}=1\implies\lambda={2\over3}\\
a=b=c={1\over3}
\]