a-2b-3c-12-2ab-3ac-6bc-48-a-b-c-

Question Number 38391 by gunawan last updated on 25/Jun/18

a + 2 b + 3 c = 12

2 a b + 3 a c + 6 b c = 48

a + b + c = \dots

Answered by MrW3 last updated on 26/Jun/18

l e t

x = a

y = 2 b

z = 3 c

(i) \Rightarrow x + y + z = 12 \Rightarrow z = 12 - x - y

(i i) \Rightarrow x y + y z + z x = 48 \Rightarrow x y + 12 (x + y) - {(x + y)}^{2} = 48

F (x, y) = x y + 12 (x + y) - {(x + y)}^{2}

\frac{\partial F}{\partial x} = y + 12 - 2 (x + y) = 0 \Rightarrow 2 x + y = 12

\frac{\partial F}{\partial y} = x + 12 - 2 (x + y) = 0 \Rightarrow x + 2 y = 12

{\begin{cases} 2 x + y = 12 \\ x + 2 y = 12 \end{cases} \Rightarrow x = y = 4 \Rightarrow F (4, 4) = 48

\Rightarrow m a x i m u m f r o m F (x, y) i s a t x = y = 4

w h i c h i s 48, t h a t i s t o s a y

{\begin{cases} x + y + z = 12 \\ x y + y z + z x = 48 \end{cases}

h a s o n e a n d o n l y o n e s o l u t i o n :

x = y = z = 4

\Rightarrow a = 4, b = \frac{4}{2} = 2, c = \frac{4}{3}

\Rightarrow a + b + c = 4 + 2 + \frac{4}{3} = \frac{22}{3}

Commented by tanmay.chaudhury50@gmail.com last updated on 25/Jun/18

e x c e l l e n t \dots

Commented by gunawan last updated on 26/Jun/18

wow nice Sir

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