Problems and Solutions

Jai Ganesh · 2005-08-11 16:07:26

Problem # n+7

If x ∈ Real Numbers, What is the minimum value of x^x?

MathsIsFun · 2005-08-11 17:52:10

Minimum?

0^0 = undefined
0.1^0.1 = 0.794
0.5^0.5 = 0.707
0.7^0.7 = 0.779

Somewhere between 0.1 and 0.7 ... I will leave the rest up to someone else

mathsyperson · 2005-08-11 20:37:01

The 'proper' way to solve this would be to find turning points by differentiating.
y=x^x.
dy/dx = x^x * ln x (I make no guarantee that that is right )
Turning points are horizontal, so we need to solve x^x * ln x = 0.
x^x * ln x = 0 when either x^x or ln x = 0.
Do ln x first, as it's easier.
ln x = 0
Take to the eth power: x=e^0=1. So 1 is a turning point, but as x^x=1, and there are values of x^x less than 1, this is clearly the wrong one.

So we are left with x^x=0. The solution to this is the minimum value of x^x. And, because of a loophole in your question, I can answer that the minimum value of x^x is 0. I don't know what value of x gives that, but you didn't ask for it .

wcy · 2005-08-11 21:01:53

for quadratic equations, you can make use of the symmetry of the graphs to find the minimum point, which is halfway in between the two x-intercepts

Jai Ganesh · 2005-08-11 21:12:32

Am I allowed to slightly modify my question?
If yes, the question should read
'For what value of x is x^x the minimum, x>0?'
If, as I had said earlier, x∈ Real numbers,
my question would have no answer as
(-1 x 10^n) -1 would give a negative odd number for a
large value of n (n ∈ Natural Numbers). And this number raised to itself would give a negative number which would be much lesser than zero!
Therefore, the smallest value would be -∞

mathsyperson · 2005-08-11 21:53:46

If you are allowed negative values, then they would have to be odd integers or reciprocals of odd integers, otherwise x^x would be imaginary or positive. The lowest value of x^x would be when x= -1/(2n+1), such that 1/(2n+1) is the closest possible to the value of x that gives the lowest value of x^x. This is -1/3, that returns a value of -1.44224927...

Large negative numbers would return values that were extremely close to 0.

Back to your original question, I brute-forced it in Excel for a while, and it came up with x≈0.3678, meaning x^x≈0.692200633.

Jai Ganesh · 2005-08-11 22:23:33

Oops, I made a mistake.

If, as I had said earlier, x∈ Real numbers,
my question would have no answer as
(-1 x 10^n) -1 would give a negative odd number for a
large value of n (n ∈ Natural Numbers). And this number raised to itself would give a negative number which would be much lesser than zero!
Therefore, the smallest value would be -∞

I forgot while posting that one that a^-n = 1/a^n
Therefore, the minmum value of x^x is not -∞
I am sorry I wasn't concentrating 100%

Jai Ganesh · 2005-08-11 22:28:30

Mathsy, you are right.
But, this can be done with Calculus.
Let x^x = y
x logx = log y
Differentiating both side,
xLogx(1) + x(1/x) = d(logy)
logx + 1 = d(logy)
When this is equal to zero,
Logx + 1 = 0
Log x = -1
x = e^-1 = 1/e
This is because logy is minimum when y is minmum

Last edited by Jai Ganesh (2005-08-11 22:29:35)

wcy · 2005-08-12 22:36:09

how to differentiate log x ?

Jai Ganesh · 2005-08-12 23:44:28

d/dx (logx) = 1/x and ∫(1/x)dx = logx

Jai Ganesh · 2005-08-12 23:53:01

Problem # n+8

Can you find two numbers composed of only ones which give the same result by addition and multiplication?

mathsyperson · 2005-08-13 00:07:36

I use 10 ones to make these: -- --
2+2=2x2. | |
Alright, I know that's not -- --
allowed, but I'm stumped | |
otherwise. -- --

NIH · 2005-08-13 11:08:24

wcy wrote:

how to differentiate log x ?

Out of interest, you can derive it from the derivative of the exponential function:
y = log x <=> x = e^y.
So dx/dy = e^y = x.
Hence dy/dx = 1 / dx/xy = 1/x.

NIH · 2005-08-13 11:18:15

ganesh wrote:

Problem # n+8
Can you find two numbers composed of only ones which give the same result by addition and multiplication?

How about

and . Works in any number base!

Jai Ganesh · 2005-08-14 15:28:53

You are correct, NIH !

Problem # n+8

Find the next term of the series
3, 14, 39, 84, 155, _____

MathsIsFun · 2005-08-14 18:31:26

Well, at least this one goes UP (rather than those up-down-left-right-next-time-add-subtract-multiply-divide types!)

3+11=14, 14+25=39, 39+45=84, 84+71=155 ... hmm ... * walks off mystified *

mathsyperson · 2005-08-14 19:51:12

3 14 39 84 155
11 25 45 71
14 20 26
6 6 ---> 6/3!=1 ---> (1)n³+...

3-1³ 14-2³ 39-3³ 84-4³ 155-5³
2 6 12 20 30
4 6 8 10
2 2 2 ---> 2/2!=1 ---> n³+(1)n²+...

2-1² 6-2² 12-3² 20-4² 30-5²
1 2 3 4 5
1 1 1 1 ---> 1/1!=1 ---> n³+n²+n+...

1-1 2-2 3-3 4-4 5-5
0 0 0 0 0 ---> 0/0!=0 ---> n³+n²+n+0n° = n³+n²+n

So the next number would be 6³+6²+6=258

Yay for brute force!

mathsyperson · 2005-08-15 11:12:41

For the one where you have two numbers that add and multiply to give the same thing, I got bored so worked out that if you are given a value of z, then you can work out values of x and y such that x+y=xy=z like so:

x = √(z(z/4-1))+z/2
y = z/(√(z(z/4-1))+z/2)

This will work for any value of z, but if 0<z<4 then x and y will be complex.

Jai Ganesh · 2005-08-15 16:10:51

mathsyperson wrote:

So the next number would be 6³+6²+6=258

Mathsy, thats very good!

Problem #n+9

From a well shuffled pack of 52 cards, four cards are drawn one after the other. What's the probability that the four of them would belong to different suits?

Last edited by Jai Ganesh (2005-08-15 16:12:15)

mathsyperson · 2005-08-15 20:17:01

Jai Ganesh · 2005-08-16 16:27:32

Correct, Mathsy!

Problem # n+10

Mathsyperson, Kylekatarn and wcy are a team for solving problems. The possibility of one solving a problem is 45%, the other is 50% and the third is 55%. What is the probability that a problem would be solved?

mathsyperson · 2005-08-16 20:23:57

This is all in my head, so there might be a mistake somewhere...
Person 1 has a 45% chance of doing it, meaning he has a 55% chance of not doing it, which is 11/20.
Similarly, person 2's chance is 1/2 and person 3's chance is 9/20.

So the chance that the puzzle won't get solved is 11/20x1/2x9/20=99/800, meaning the chance that it will is 701/800.
Convert back into a percentage: (701/8)% =87.625%

It's probably less than that though, because we would all solve easy problems and all not solve difficult problems. This assumes that we solve problems independant of each other.

Jai Ganesh · 2005-08-17 16:29:10

Problem # n+11

A circle of diameter 'a' is cut from a square metal sheet of side 'a'. From the four corner cuttings, four circles are cut. Whats the maximum area of the four circles put together?

Jai Ganesh · 2005-08-22 16:25:14

Problem #k
What is the total number of squares, of all sizes, in a Chess board?

mathsyperson · 2005-08-22 22:40:32

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