<s>[INST] To find maximum value when sum is fixed Given that $a,b,c,d,e$ and $f$ are six non-negative real numbers such that
$a+b+c+d+e+f=1$.
Find the maximum value of
$ab+bc+cd+de+ef$.
My Approach
After lot of thinking I reduced the problem to following:
Given that $a,b,c,d,e$ and $f$ are six non-negative real numbers such that
$a+b+c=1$.
Find the maximum value of
$ab+bc$.
which is quite easy to solve:
i.e.  $ab+bc=b(a+c)=b(1-b)=\frac{1}{4}-\left(b-\frac{1}{2}\right)^2\leq\frac{1}{4}$.
Not able to extrapolate to the given problem in hand.
Is there a Calculus approach.
 [/INST] Let $0\leq a \leq b \leq c \leq d \leq f \leq e\leq 1$
$$a+b+c+d+f+e=1 \rightarrow (a+b+c+d+f+e-1)^2=0$$ 
$$\rightarrow a^2+b^2+c^2+d^2+f^2+(e-1)^2+....=0$$
so we have $a^2=b^2=c^2=d^2=f^2=(e-1)^2=0$ i.e., $a=b=c=d=f=0$ and $e=1$.
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