Hi, there!
Bravo to a Japanese mathematician who seems to have provided
a resolution of the ABC conjecture.
An old cliché is quoted as saying that good deeds are contagious.
So such a Japanese high-caliber mathematician has a good ripple
effect on a tenacious and ambitious researcher to materialize an
assignment.
I am not that studious, but I have room to enjoy estimating maxim-
um utility in the word problem described below.
Maximize utility \(u = f(x, y) = xy\) subject to the constraint \(g(x, y)\)
\(= 2x + 3y = 560\). Here the price of per unit \(x\) is 2, the price of \(y\) is
4 and the budget available to buy \(x\) and \(y\) is 560. Utility may be ma-
ximized at (***, ***).
The optimization problem comes down to:
Objective function: maximize \(u(x, y) = xy\)
Following the constraint: \(g(x, y) =2x + 3y = 560\)
First, \(－\frac{fx}{fy} = －\frac{y}{x}\), which shows the slope of the indifference curve.
Second, \(－\frac{gx}{gy} = －\frac{1}{2}\), which represents the slope of the budget line.
Third, \(－\frac{fx}{fy} = －\frac{gx}{gy}\); the utility maximization requires the slope of
the indifference curve to be equal to the slope of the budget line.
\(－\frac{y}{x} = －\frac{1}{2}\), then \(x = 2y\)
Fourth, from the third step, use the relation between \(x\) and \(y\) in the
constraint function to get the critical values.
\(2x + 3y = 560\), then \(4y + 3y = 560\), so \(7y = 560\), \(y = 80\)
Using \(y = 80\) in the relation \(x = 2y\), we get \(x = 2✕80 = 160\)
Utility may be maximized at (160, 80).
I’ve seen a similar problem on another website, the source of which
I totally forgot. Sorry. Once someone has identified the exact resour-
ce, I will be more than happy to quote the specific site which I elicited
the problem from.
To be a mathematics expert is still far-reaching, but I stay tenacious
to be as close to that stage as possible.
Stay tuned, and expect to see my next post.
Keep well.
Frank Yoshida
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【グローバルビジネスで役立つ数学】でもっと学習する b＾＾)
【参考図書】『もう一度高校数学』（著者：高橋一雄氏）株式会社日本実業出版社
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