Problem # n+7

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

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 .

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 -∞

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%

how to differentiate log x ?

Problem # n+8

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

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.

You are correct, NIH !

Problem # n+8

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

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 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 ganesh (2005-08-16 14:12:15)

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?

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?

here it is... just the answer

Last edited by kylekatarn (2005-08-23 14:41:39)