Why is this?

[lrdheat]

Likely simple math, but I will ask anyway…I was messing around with graphing various functions. When I graphed x*e^-(x^2), I discovered that the max was equal to sin (45 degrees). Why?

[Valentin Albillo]

(04-07-2022 02:12 AM)lrdheat Wrote:  Likely simple math, but I will ask anyway…I was messing around with graphing various functions. When I graphed x*e^-(x^2), I discovered that the max was equal to sin (45 degrees). Why?

Sin(45^o) simply equals Sqrt(2)/2, which holds no mystery whatsoever.

V.

[Leonid]

(04-07-2022 02:12 AM)lrdheat Wrote:  Likely simple math, but I will ask anyway…I was messing around with graphing various functions. When I graphed x*e^-(x^2), I discovered that the max was equal to sin (45 degrees). Why?

By the way, global max \[\sqrt[x]{x}\] is e

[lrdheat]

Yes, and this is more intuitive to me. Is it purely coincidental that the max
of x*e^(-x^2) is (sqrt 2)/2, or is there a direct relationship to the sine function? I’m thinking that I am missing some basic math…

[Albert Chan]

f = x * exp(-x^2) is product of linear growth and exponential decay. A peak is expected.
Peak of f also at peak of ln(f) = ln(x) - x^2

(ln(f))' = 1/x - 2x = 0, we have x^2 = 1/2, or x = √(1/2)

---

g = x^(1/x) = exp(ln(x)/x)
ln(x) does not grow as fast as x, so again g is decaying after a peak.
Peak of g also at peak of ln(g) = ln(x)/x

(ln(g))' = (x*(1/x) - 1*ln(x)) / x^2 = (1 - ln(x)) / x^2 = 0, we have ln(x) = 1, or x = e

[Valentin Albillo]

(04-08-2022 01:40 AM)lrdheat Wrote:  Yes, and this is more intuitive to me. Is it purely coincidental that the max
of x*e^(-x^2) is (sqrt 2)/2, or is there a direct relationship to the sine function? I’m thinking that I am missing some basic math…

- The derivative of y = x * e^(-x^2) is y' = e^(-x^2) * (1 - 2 * x^2)

- To find the extrema of y you must solve y' = 0.

- The exponential term is never 0 and the polynomial term 1 - 2 * x^2 is 0 for x = Sqrt(2) / 2, which is where y has a maximum.

- There's no relationship whatsoever to the sine function, just numerical coincidence with the numerical value of sin(45 deg), that's all.

V.

[lrdheat]

Yes, it makes sense as far as the idea of an increase and then a decay in these functions. That the max of x*e^(-x^2) occurred at x=(sqrt 2)/2 suggested a trigonometric relationship to me, but how? It is beginning to seem coincidental now, and there is not a direct trigonometric relationship…