+0  
 
+2
50
4
avatar+121 

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
avatar+9310 
+4

\(a + b = 9\), 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
avatar+9310 
+3

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
avatar+2401 
+2

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
avatar+117116 
+2

Great work Catmg and Max :)

Melody  May 4, 2022

6 Online Users