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# Problem of Tartaglia (1500-1557): among all positive numbers a, b whose sum is 9, find those for which the product of the two numbers...

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Problem of Tartaglia (1500-1557): among all positive numbers $$a$$$$b$$ whose sum is 9, find those for which the product of the two numbers and their difference is largest. (Hint: Let $$x = a - b$$ and express $$abx$$ in terms of $$x$$ alone.)

$$a =$$ ?

$$b=$$ ?

May 3, 2022

#1
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$$a + b = 9$$, so $$b = 9 - a$$.

Now, $$ab(a - b) = a(9 - a)(a - (9- a)) = a(9 - a)(2a - 9) = -2a^3 + 27a^2 - 81a$$.

When ab(a - b) is the largest, d(ab(a - b))/da = 0.

$$\dfrac{d}{da} (-2a^3 + 27a^2 - 81a) = 0\\ -6a^2 + 54a - 81 = 0\\ 2a^2 - 18a + 27 = 0\\ a = \dfrac{9 \pm 3 \sqrt 3}2$$

For brevity, let $$f(a) = -2a^3 + 27a^2 - 81a$$

$$f''\left(\dfrac{9 + 3\sqrt 3}2\right) = -12\left(\dfrac{9 + 3\sqrt 3}2\right) + 54 = -18\sqrt3 < 0$$

$$f''\left(\dfrac{9 - 3\sqrt 3}2\right) = -12\left(\dfrac{9 - 3\sqrt 3}2\right) + 54 = 18\sqrt3 > 0$$

So, the maximum value of ab(a - b) is attained at $$a = \dfrac{9 + 3\sqrt3}2$$.

Since b = 9 - a, $$b = \dfrac{9 - 3\sqrt3}2$$ when ab(a - b) is the largest.

Therefore, $$a = \dfrac{9 + 3\sqrt3}2$$, and $$b = \dfrac{9 - 3\sqrt3}2$$

May 3, 2022
#2
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By the way, I would love to see a solution without using calculus, since calculus is invented after Tartaglia.

MaxWong  May 3, 2022
edited by MaxWong  May 3, 2022
#3
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Your answer looks very cool, but I wanted to try to see if it can be solved with calculus too because I haven't learned it yet. :))

m = maximum of ab(a-b)

x = a - b (like the hint said)

9 = a + b

a = (9+x)/2

b = (9-x)/2

m = maximum of (9+x)/2*(9-x)/2*x

Note: x has to be be nonegative since if b > a, and a and b are positive integers, ab(a-b) will be negative.

This is a cubic with roots at -9, 0, and 9.

By plugging in points and drawing a sketch, we can get an idea of what the cubic looks like.

The graph approaches 0 from the positive side from as x approaches -9 from the negative side.

Then, the graph crosses the x axis at -9, and goes down, but then goes back up to the x axis at 0.

After that, the graph dips up but returns back to 0 at x = 9, and just continues going down.

So our maximum is at the turning point between 0 and 9, this is the distance m.

Then we can use the method written here:

https://www.themathdoctors.org/max-and-min-of-a-cubic-without-calculus/

=^._.^=

May 3, 2022
#4
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Great work Catmg and Max :)

Melody  May 4, 2022