Find-maximum-value-of-the-product-xy-72-3x-4y-for-positive-value-of-x-amp-y-

Question Number 141164 by iloveisrael last updated on 16/May/21

F i n d m a x i m u m v a l u e

o f t h e p r o d u c t x y (72 - 3 x - 4 y)

f o r p o s i t i v e v a l u e o f x & y .

Commented by mitica last updated on 16/May/21

m a x = 1152 f o r x = 8, y = 6

Answered by EDWIN88 last updated on 16/May/21

The terms of the product x, y & 72 - 3 x - 4 y

do not have a constant sum . However if we

adjust the product and write it as

\frac{1}{12} (3 x) (4 y) (72 - 3 x - 4 y)

then ignoring the multiplier \frac{1}{12} for a moment

we see that the factors 3 x, 4 y & 72 - 3 x - 4 y

have a constant 72 . Thus we seek value of

x & y satisfying

3 x = 4 y = 72 - 3 x - 4 y

given the unique values {\begin{cases} x = 8 \\ y = 6 \end{cases}

Thus the \max value of xy (72 - 3 x - 4 y) is

8 \times 6 \times 24 = 1, 152 ⋇

Commented by iloveisrael last updated on 16/May/21

Y e s \dots .

Answered by mitica last updated on 16/May/21

i f 72 - 3 x - 4 y ⩽ 0 \Rightarrow x y (72 - 3 x - 4 y) ⩽ 0

i f 72 - 3 x - 4 y > 0 \Rightarrow \frac{1}{12} \cdot 3 x \cdot 4 y \cdot (72 - 3 x - 4 y) \overset{a m - g m}{⩽} \frac{1}{12} {(3 x + 4 y - 72 - 3 x - 4 y)}^{3} \cdot \frac{1}{27} = 1152

= f o r 3 x = 4 y = 72 - 3 x - 4 y \Rightarrow x = 8, y = 6

\Rightarrow m a x {x y (72 - 3 x - 4 y)} = 1152

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