If-a-b-c-4-then-find-a-3-b-3-c-3-

Question Number 60484 by Askash last updated on 21/May/19

I f a + b + c = 4

t h e n f i n d

a^{3} + b^{3} + c^{3} = ?

Commented by MJS last updated on 21/May/19

{\begin{cases} a + b + c = 4 \\ a^{3} + b^{3} + c^{3} = x \end{cases}

2 equations in 4 unknown \Rightarrow parametric

form with 2 parameters

c = 4 - a - b

x = a^{3} + b^{3} + {(4 - a - b)}^{3} with a, b \in C

{that}^{'} s the perfect answer

Commented by MJS last updated on 21/May/19

we get a unique solution only if a, b, c \in N^{★}

\Rightarrow a = 2 \land b = c = 1 \lor b = 2 \land a = c = 1 \lor c = 2 \land a = b = 1

\Rightarrow a^{3} + b^{3} + c^{3} = 10

Answered by tanmay last updated on 21/May/19

\frac{a + b + c}{3} ⩾ {(a b c)}^{\frac{1}{3}}

\frac{64}{27} ⩾ (a b c)

\frac{a^{3} + b^{3} + c^{3}}{3} ⩾ {(a^{3} b^{3} c^{3})}^{\frac{1}{3}}

a^{3} + b^{3} + c^{3} ⩾ 3 \times a b c

i f w e c o n s i d e r a b c = \frac{64}{27}

t h e n a^{3} + b^{3} + c^{3} ⩾ 3 \times \frac{64}{27}

a^{3} + b^{3} + c^{3} ⩾ \frac{64}{9}

Commented by Askash last updated on 21/May/19

I w a n t p e r f e c t a n s w e r .

Commented by mr W last updated on 21/May/19

h o w c a n y o u s a y A ⩾ C f r o m

A ⩾ B a n d B ⩽ C ?

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