Question

 

If a; b; c are positive real numbers satisfying ab+bc+ca = 12, then the

maximum value of abc is

Answer 1

The expression is completely symmetrical in a, b and c, so let ab=bc=ac, and a=b, then 3a^2=12 and a=b=c=2, and abc=8.

Let a=2-x and b=2+x, c=(12-ab)/(a+b). Since c>0 ab<12. c=(12-4+x^2)/4=(8+x^2)/4=2+x^2/4.

abc=(4-x^2)(8+x^2)/4=(32-4x^2-x^4)/4=8-x^2-x^2/4. When x=0 abc=8 and a=b=c=2. For any other values of x, abc<8, therefore the maximum value of abc is 8.