If $\displaystyle a,b,c$ are positive real numbers such that $\displaystyle a+b+c=1$ and
$\displaystyle \lambda=min\{a^3+a^2bc,b^3+ab^2c,c^3+abc^2\},$then find the value of $\displaystyle \lambda$
I would like to know if this is the question flat out or if this question arose from another question...
Without loss of generality, we can consider only $\displaystyle a^3+a^2bc$ since the other two expressions are the same but with a=b and a=c, because there is no reason why you can't call the 3 numbers a,b,c in any order when you find them
Now, if we let a approach zero, the expression approaches zero. From what you've said a cannot be 0, but you can let a keep getting smaller and the value of lamda will continue to decrease
For instance, let b=.49998 and c=.49998 and see what happens
Then let b=.4999999998 and c=.4999999998 and see what happens (with a calculator you'll probably get an answer of 0 with round off)