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Question Number 62176 by Tawa1 last updated on 16/Jun/19

	Answered by MJS last updated on 17/Jun/19

	$to make it clear let : a, b, c, d \in R$ $a + b + c + d = 1$ $F = a b + b c + c d$ $d = 1 - a - b - c$ $F = a b + c - a c - c^{2}$ $\frac{d F}{d a} = b - c = 0 \Rightarrow c = b$ $\frac{d F}{d b} = a = 0 \Rightarrow a = 0$ $\frac{d F}{d c} = 1 - a - 2 c = 0 \Rightarrow c = \frac{1 - a}{2} \Rightarrow c = \frac{1}{2}$ $\Rightarrow b = \frac{1}{2} \Rightarrow d = 0$ $F = a b + b c + c d = \frac{1}{4}$ $we get the same on a different path$ $c = 1 - a - b - d$ $F = (b + d) - {(b + d)}^{2} - a d$ $\frac{d F}{d a} = - d = 0 \Rightarrow d = 0$ $\frac{d F}{d b} = 1 - 2 (b + d) = 0 \Rightarrow b = \frac{1}{2} - d \Rightarrow b = \frac{1}{2}$ $\frac{d F}{d d} = 1 - 2 (b + d) - a = 0 \Rightarrow a = 0$ $\Rightarrow c = \frac{1}{2}$ $this means : b = c must be as big and a = d as$ $small as possible$ $in our case$ $a, b, c, d \in N^{★}$ $a + b + c + d = 63$ $a = d = 1$ $b = 31 \land c = 30 \lor b = 30 \land c = 31$ $F = 1 \times 31 + 31 \times 30 + 30 \times 1 = 991$ $if a, b, c, d \in R$ $a = d = 0$ $b = c = 31.5$ $F = 31 . 5^{2} = 992.25$
	Commented by Tawa1 last updated on 21/Jun/19

	$God bless you sir$
	Answered by mr W last updated on 21/Jun/19

	$a + b + c + d = 63$ $S = a b + b c + c d$ $= a b + b c + c d + d a - d a$ $= (a + c) (b + d) - d a$ $= (a + c) [63 - (a + c)] - d a$ $t o g e t m a x . S, d m u s t b e a s s m a l l a s$ $p o s s i b l e w h i c h i s 1 .$ $\Rightarrow a + c = 62 - b$ $S = (62 - b) (1 + b) - a$ $t o g e t m a x . S, a m u s t b e a s s m a l l a s$ $p o s s i b l e w h i c h i s 1 .$ $\Rightarrow b + c = 61 \Rightarrow 1 ⩽ b ⩽ 60$ $S = (62 - b) (1 + b) - 1 = 61 + {(\frac{61}{2})}^{2} - {(b - \frac{61}{2})}^{2}$ $t o g e t m a x . S, b s h o u l d b e a s c l o s e$ $t o \frac{61}{2} a s p o s s i b l e, w h i c h i s 30 o r 31 .$ $w i t h b = 30 : S = 32 \times 31 - 1 = 991$ $w i t h b = 31 : S = 31 \times 32 - 1 = 991$ $\Rightarrow m a x . S = 991$
	Commented by Tawa1 last updated on 21/Jun/19

	$God bless you sir$
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